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If China would never allow it, let's just assume they aren't a factor and NK stands alone.
Would the SK government even be willing to take a completely broken country under its wing? And if they did could they ever hope to establish a successful economy there?
Would other countries like the EU or America be willing to financially support such a project and is there even something to be gained from a unified Korea from a western perspective? What would be some of the biggest roadblocks?
Disclaimer: counterfactuals are hard to answer period and I'm certainly not a professional - but I'll give a crack at it!
The starting context you need here is that Korea being split is a modern phenomenon and a historic aberration. Imagine it: the Asian peninsula was mostly left alone till the 19th century and then in a period of around 60 years, Korea was conquered and raped (I use that word intentionally - I think it's a fair characterization of occupied Korea's treatment) by the Japanese then sliced up in two by the two Cold War great powers. The traditional folk song Arirang is popular in both the North and South because of the great longing the song has within it.
The tl;dr of that is that South Korea would absolutely want to reunify even if it didn't make economic or tactical sense. As it is, China would probably take charge of North Korea if the South Koreans (backed by the US) didn't so they have a strong tactical reason to want to control it. South Korea actually has a government agency called the Ministry of Reunification that advocate for and prepares for reunification.
Something to note here is that prior to the Kims taking over North Korea and driving it into a Banana Republic, North Korea was wealthier than South Korea. That's because it naturally has more arable land and useful natural resources. South Korea being an OECD country is a developmental miracle (particularly when you consider that countries starting at a similar point like India didn't make anything close to the same gains).
All that said, changing North Korea would be hard. First of all, you'd have to cope with the obvious issues of pro-NK terrorists hiding within the populace. Then you'd have to address the massive poverty and famine issues (although this has been improving in recent years). Then you'd have to start imposing a political system on them that can respect property rights and encourage development. In between all those high level goals is a million tiny things that can go wrong (and this is all assuming a good scenario where China doesn't send its troops in, where all the nuclear weapons are successfully recovered, and this doesn't happen after a devastating war). I suggest reading the RAND report above for more details but one thing is clear: the unified Korea's GDP would likely be lower in the short term than their current sum of GDPs.
Let's assume that we're in a world with mostly standard Western foreign policy. This basically assumes that Trump or a Trump-like figure is not in charge of the US and that the EU is still relatively stable.
The West would absolutely be involved. First of all, the US has a massive troop presence there basically to prepare for something like this. Secondly, the West actually has a pretty good track record of humanitarian involvement and this would be a massive project. Finally, they would want to reduce Chinese involvement to whatever extent possible.
Significantly, South Korea is also reducing its military size right now out of population concerns. They would absolutely need American forces to hold North Korea and distribute the food and medicine needed.
Anna Fifield's excellent book on Kim Jong Un
Victor Cha's The Impossible State
https://www.rand.org/content/dam/rand/pubs/research_reports/RR300/RR331/RAND_RR331.pdf
