First of all a lot of the popular support for "socialism" in the U.S. is probably a backlash from the conservative fearmongering, which for decades called anything left of anarcho-capitalism the dreaded "s"-word, so much so that people are starting to use it again, though largely referring to social-democratic (social-market-economic) policies rather than actual socialism.
With regards to Marx and Marxism. Afaik Marx himself didn't actually contribute all that much to the question how a communist society should look like and how it's supposed to be achieved or organized. He was active in the 1st international (workingmens association) and bullied the anarchists out of it and his communist manifesto included a 10 point action plan which he called largely obsolete in the preface of later versions. He also vaguely spoke about a "dictatorship of the proletariat" and a transitional period between capitalism and communism and approved of revolutionary terror to make the process go a faster. Though contemporaries and later theoreticians have placed way more value on these terms than Marx has. He wasn't too active in the revolutions in his lifetime an the ones he saw in 1848 and the Paris Commune in 1871 were short lived and rolled back pretty fast.
So the idea that he developed a rudimentary masterplan for how society should be organized is largely false. His contribution is apparently the description, analysis and critique of the upcoming industrial capitalism of his time. He dipped his toes into economic and social science developing a theory of value based on Smith and Ricardo and took on a materialistic perspective towards social relations. So the broad social structure of classes, castes and hierarchies is based on economic relations and material conditions. He even conjectured a historic materialism where he views history through the lens of class struggles.
Now he was hardly the first to recognize that there is a discrepancy of power between those who own the mean of production and those who work them or to call for various ways to get rid of that discrepancy. Though he was more interested in making it a science while others were apparently more interested in describing their utopia or engaged in practical activism.
Apparently Marx is a seminal figure for many branches in economics, sociology and philosophy and "Marxism" thus can refer to a variety of ideas going back to Marx. Also even during his lifetime Marx had his struggle with "Marxists" as interpretations of his works didn't always match with his own interpretations.
With regards to democracy. Well the ideal, or the direction of the historic progression seen by Marx is democratic. To him these class struggles present themselves as a movement from the rule of the few to the rule of the many or an anarchist scenario without rulers. And with the industrialization and the increase in economic production, the relations between the classes in this productive system are no longer a necessity of the production process but are merely kept in place by the ownership of the means of production, which is a title granted and protected by an oppressive state. But the proletariat is, or is to become so plentiful and powerful that they could do away with that and enter a new stage with new problems.
To an extend he even sees capitalism as necessary and positive in doing away with middle classes and religious, ethnic and other differences establishing only class as the relevant distinction that creates an ever growing proletariat, that if it would recognize it's common socio-economical situation would just need to join forces and organize and would be basically unstoppable given the numbers and the improved material conditions. So when talking about a communist party he's talking about a mass movement, not a top-down vanguard party. Afaik Engels even criticized that approach after the Paris Commune as failure.
The mode of change is a revolution. Now for context, he apparently wasn't generally an adversary of peaceful transitions but if you take a look at the political landscape of his time you'd find that democracies where something rather scarce. France pivoted between a republic and a absolute monarchy and the U.S. and England which were nominally democratic or at least technically democratic constitutional monarchies took well into the 20th century to establish even universal suffrage.
The founding slave owners of the U.S. were so contemptuous of the liberty of the working class that they called democracy "tyranny of the majority" and "mob rule". In Federalist Papers No. 10 Madison even shared Marx's impression (not by name but by idea) that the working class would form factions based on their shared economic situation and demand change, just that unlike Marx he doesn't see that as something good, but instead specifically argues for the state to uphold the privileges of the better offs and to find ways to curb the ability of the rest to resist that, which famously includes his OPPOSITION to DEMOCRACY. While in Marx's exile in London also only a small fraction of the population was eligible to vote with the poor and working class being among those without suffrage.
Meanwhile on the European continent revolutions for democracy were militarily crushed and there was a period where monarchy had a comeback. So even the phony or shortlasting democracies had to be fought for by revolutionary violence and actual democracies were as utopian as socialism. So given these circumstances it's somewhat understandable that Marx as well as many others was primarily in favor of a revolution as reforms seemed to be fully out of reach, he made exceptions though for countries that already had nominal democracies and the ability to do it peacefully and apparently argued in 1952 that the universal suffrage in England would be the most socialist thing to happen compared to anything happening on the continent, because given that the proletariat has the majority it would be equivalent to the rule of the proletariat.
Now obviously history proved many of these assumptions wrong, insufficient, not yet realized or anything else. Capitalism didn't crush all differences between people except class, in fact fascists and conservatives still pit different sections of the poor and working class against each other, by "perceived class", sex, race, ethnicity and so on. Nationalism proved to be more successful than anticipated in making people believe it's "their country" despite not owning even a handful of dirt of it. He was kinda right that revolutions were in the air, as can be seen in the beginning of the 20th century, though again quite some of them were unsuccessful and as a response to the successful ones the ruling classes rather reformed than risked less structured transitions. Democracy, unions and social security further reduced the necessity for revolutions, both in terms of material necessity as well as in the sense of political means. Which depending on perspective can be seen as progress being achieved or progress being inhibited as a marginally better status quo is frozen rather than advanced further. Also universal suffrage didn't lead to large scale victories of socialist mass movements and despite being a majority the concerns of the lower classes are still only of marginal interest to the plurality of parties despite their perceived plurality.
Despite already being debunked with the enlightenment, the idea of an aristocracy (rule of the best) or it's modern version a "meritocracy" are still being floated, as well as a pseudo-religious prosperity gospel and a cult of personality around rich people, despite better knowledge. Technically powerful techniques such as general strikes and boycotts are often either undermined by individuals betrayal or outlawed and branded non-effective.
Though whether that's the end of history is not yet clear, given that capitalism still suffers from the same problems already described by Marx, it's still a system that alienates people from their labor, it amasses riches with the few and impoverishes the many (if left unregulated). And while democracies regulated a lot of the worst aspects of it, there's a constant struggle of those profiting from exploitation to deregulate them again. Or urgent patch regulations can end up granting privileges to certain groups. The fact that staggering economic inequalities endanger the nationalist narrative and further the social class perception of a majority of people. So that just be wrong in the sense of an inevitable linear progression but not in principle.
On the other hand Lenin apparently saw a whole lot of that already at the start of the 20th century while being exiled in western Europe and despite the economic conditions being ripe for change, no such change occurred revolutionary.
From which he again went back to the vanguard party, the dictatorship of the proletariat, not by majority but by seizing state level powers and by forcing a lot of the collectivist ideas. Which has many problems. Like capitalists like to point to how mean revolutions are and that they weren't as economically successful as countries not just emerging colonialism and banana republics or having "won" (lost) a devastating war. But lots of countries had revolutions, the problem is rather that it's a minority revolution, so more of a coup, which means they have a majority against them which has a much higher likelihood of going dictatorial to preserve power, while a majority movement would not need to be as threatened by a minority.
Also if this is supposed to be a transitional period, where is the transition? I mean democracy is a process that people need to learn and how do you learn it if you can't effectively make decisions (as a group)? But are given orders top-down? How do you overcome capitalism if you still engage in it, still produce for sale rather than consumption with the revenue of the trade going to the "owners" or in that case managers of the means of production rather than the workers?
TL;DR Marx didn't really produce a theory of socialism/communism that could be the foundation of a democratic regime, he criticized capitalism and had ideas of a historic trend. Of which he was likely wrong.
Though is it fundamentally impossible for a mass movement to develop class consciousness, seize the productive economy and restructure it so that everyone is worker and owner and the productive output is produced, invested or distributed democratically? No not at all, parts of the state and the economy already function like that, democracy has gone a long way since Marx was around and it's technically thinkable. Whether that works or has new and different problems, some of which we've already seen or am seeing, is a different question that goes beyond Marx and his writings anyway.