One problem with your definition of "left wing" and "right wing" parties over whom they benefit is with the ambiguity of what "benefits" mean. Like you seem to think of that purely in monetary terms, but historically and how it's apparently used in social sciences: it's about who has the power to determine that path of society.
Where right wing parties are usually elitist, favor strong leadership, backdoor deals and "government as a service" or more drastically who see the citizenry as customers who are exploited for their workforce, creativity, strength and whatnot and are entertained to keep them available for the good of "the state" (and in consequence the states leaders).
While left wingers are meant to be more democratic. Who reject strong leadership, like to incorporate more people into the decision making process, try to increase general education and ability to take part in the political process and who see government as the self-organization of the people, by the people for the people. So "the government" doesn't provide a service, the government are the people who decide for themselves how they want to progress as a society (at least in theory).
So for the left "the poor" are part of the people and citizen in distress who should be assisted, while for the right "the poor" are somewhere between uninteresting customers (no money to pay for services) or a nuisance because they need assistance and provide (seemingly no benefit). Whether this idea on the right is correct to begin with is already dubious given that just because people don't make a lot of money doesn't mean they work less or conversely just because people make a lot of money it doesn't mean they work a lot, but on paper the poor are in need of money rather than being the source of it so from the perspective of the (established) right they are usually treated as disposable.
And then there is a phenomena called "populist parties" who can be either left or right wing (though in effect they are or end up being right wing), who see the poor as mostly instrumental in getting into power, but just like an instrument they think of them as a means to an end not as an end in itself. So idk your fascists, racists, xenophobes etc who pretend that "you" as a poor guy phenotypically or socially part of a majority group are actually part of the team and the only reason you are so downtrodden is because of [insert scapegoat minority]. So instead of outright promising a better life they promise to go after [insert scapegoat minority]. Which either doesn't work because there are pesky things such as LAWS and HUMAN RIGHTS... Which only makes their call appear more urgent because their obvious solution is "unjustly ignored" by "the elites". Or which if it does work and they do what they promised, "miraculously" does nothing for the poor because more likely than not [insert scapegoat minority] was not the mastermind behind everything evil in the world (who could have seen that coming...). So while on the face of it talking to the poor, their politics is generally not geared towards the poor or even the majority of the people but rather at getting their support or tolerance of their politics with passion and emotionalized rhetoric of "us vs them" ("build the wall", "lock em up", "illegal aliens", etc) rather than with any concrete political proposal as long as they don't have the power yet to not give a shit about them. The other side effect is that if the different groups of disenfranchised minorities hate upon each other, they can't join forces and demand better from the rulers and if they really hate each other, then you can sell a different brand of the same bullshit to all of them because they don't talk to each other about it and don't realize they're getting played.
Now the relationship between populist parties and the established right wing is complicated. On the on hand the old right doesn't particularly like the alt right, they are often loud annoying, bad for business and draw too much negative attention to them, cannibalize their classical voter base and might challenge their power, on the other hand they might be popular, "inoffensive" to the goals of the old right (sure they might kill minorities but the old right are usually not persecuted minorities) and they provide a motivation and impetus to old right topics that the old right couldn't muster themselves (like they are not supposed to be look cool to "you", because "you" are not supposed to be their peer group).
So in effect, despite the differences, the old right often enough forged uneasy alliances with the alt right (or adopted these positions themselves), whether that's Italian conservatives and Mussolini, German aristocrats and Hitler or U.S. conservatives and Trump. Despite their disqualifying antics, lies, conspiracy theories and anti-democratic goals,... unlike leftists who actually ask for change, these figures promised the same as ever just with more radical rhetoric and their "enemies" overlap so they were not a threat to the establishment. They are a threat to democracy, idk stuffing supreme courts with cronies, trying to take over positions of power and ignore rule breaking of their side while punishing it for others, delegitimizing elections and searching for excuses for illegal power grabs etc, but they don't usually pretend to go against the political establishment. Like if Trump floated the idea of nationalizing key industries or raising the taxes on the rich or anything that actually disturbs the conservative worldview that would be a different questions but as long as "drain the swamp" is followed by making it more swampy they don't really give a shit.
The problem is that at some point the populist has to present results and either people realize that the king is bare naked or he'll deflect by going more forcefully against the [insert scapegoat minorities] who are behind the fact that nothing he proposed worked as expected.
On the other end leftist populist usually talk about how revolutionary and avant-garde/progressive they are which sounds like change and is a lot less accepted by the established political parties who like to keep things the way they are (namely themselves in power). So they usually don't even take up speed, but if they do and actually come close to a revolution they run into a similar problem that their simplified worldview of fighting against the elites and trying to seize power runs into the same set of problem, that they end up being the elites once they seized power and not distributed to the people. So as a result things remain as they were just differently themed and they have no means to keep the momentum (of a revolution) without force, lies and deception.
Now after all that preface where do "the rich" fall into that and why do they seemingly have better means to get what they want?
Well the rich are themselves in the center of power. Albeit not necessarily the political power but the economic power. That's why they are rich to begin with, because they hold access over key positions and command armies worth of workers. So idk if Zuckerberg makes a statement, Facebook (or Meta or whatever) makes a statement. Even if he doesn't do that as CEO, you still know that he's the guy who has effective control over ~100 billion dollar revenue, ~100,000 employees and ~3 billion reachable customers make a statement. And idk if he decides to, on a whim move the headquarter to some other place that can utterly disrupt the economic landscape in both the place that was left (negatively) and the place where they are moving to (positively).
So as the power of the political parties stems mostly from their ability to spend unbelievably large sums of tax revenue and as rich people hold the power of the sources of that revenue (not because it's some conspiracy but because holding positions of power is what makes the holders rich...), it is very likely that politicians will do "what is good for the economy" and those rich people right at the spot where "the economy" is happening more often than not end up telling them "what is good for the economy", which coincidentally might also be what is good for them personally. Is that always sound economic policy? Well ask the experts, who are they again? Oh right rich people or people employed by rich people in their lobbying institution ... I mean "think"-tanks.
Now do rich people only ever use their money, influence and reach to benefit themselves? Yes, no, maybe. Like some give away lots of money which is likewise benevolent and raises their reputation, reach and might be tax deductible. So maybe they mean it, maybe it's just PR, who knows. Also what is spending to both parties other than a "don't forget about me".
So the thing is rich people don't need to be good at detecting who's serving their interest both, they largely can serve their own interest with their positions of power in the economy and given they sit at the source of information about that they can tell politics what's best for them rather than being told the other way around.
Also rich people can afford to be individuals, they can vote for whatever they want for whatever reason they want. If Elon Musk wants to tell the world something he can simply buy RIP formerly known as twitter (or what does the X stand for?). If a poor person wants to tell the world something... well good luck. He can scream in a public place and be arrested by the police or needs to join a party or faction of any kind that multiplies his voice by having more people repeat it, though at the price of finding compromises and all the pesky democracy bits that the rich can circumvent with money and power.
Also the poor are most often overlooked by any party as their priority lies with the rich and larger or smaller sections of the middle class. As an effect of that the poor are often completely ignored and might not even vote which gives the powerful even more reason to ignore them, but even if they vote they have no means to tell the world for whom and why so they are again insignificant to the poll watchers and thus likely not accounted for. Or if they vote continuously for one party the other party counts them as lost, while the party that they vote for counts them as safe bet and again neither of the parties gives a shit.
You know other than populists, but as we've covered they give even less of a shit about them, no matter what they say.