Theoretically checking this seems bad unless you're giving up because the pot is big relative to stacks but checking and "playing poker" might have some merit.
If button bets and everyone folds you should XRAI unless button is absurdly straightforward.
If button bets and everyone calls you have an easy fold.
If button bets and gets one other call you're in judgment territory but probably XRAI. Equity against ranges is probably good.
The turn card can be either obviously bad, obviously awesome (the ace of hearts), or pretty good but might have made someone two pair or trips. Still, you'll have the same four players to act. If they bet probably fold based on reads and the specific card. If they check to you on a safe turn, hmmm.... pot and always get called by 2 pair? Check back and pray for a safe board pair?
The downside is bottom or middle pair could make trips or two pair and never give up. At least you have some equity against two pair. Naked flush and straight draws are now making a big mistake to call you. Often you'll get stacked this way but sometimes you'd have gotten stacked by betting, and at least this way you have more info.
What I just wrote feels theoretically wrong but makes up for it in creativity.