No one needs to do anything here. People can research and find stuff out for themselves, and come to their own conclusions.
Twitter can clearly not fact-check every single tweet on its platform. But what if they did it for every tweet (maybe from a verified account) that X people report for being untruthful? Trump would look bad even if Twitter held everyone to the same standard, and the blue checkmark would come with some responsibility not to lie, so why not?