You make a valid point about all factors. And you also make a good point about the texture of this board being not too scary in terms of letting free cards pop off on turn. It's not flush draw heavy, it's not really connected (although villain could have 78 or 79 and even then not too scary).
However, in considering opponents range here what might they have? We can assume they don't have AA-QQ as they're likely going to 4 bet you pre flop here anyways (although as you said, there's a lot to consider, I stacked a guy recently who thought it was a good idea to limp in with AA). But what if they do have 77 or higher or a gut shot, or 3 to a flush? Yeah not super terrifying but what do you do when the turn comes? What if it makes a flush draw more likely, or turns a gut shot into an open ended straight draw? What if they luck out and hit a higher set you gave them for free? Now you've not only given away a free card that is now dangerous, you've also missed a golden opportunity on the flop to have built a bigger pot to get them chasing even harder.
Overall giving away free cards is going to be a very situation specific maneuver and not one most players will advocate the vast majority of the time. By check-calling the flop here you 1.) lose potential value of a larger pot you could be building and 2.) give yourself a harder decision on the turn. A lot of players at 1/2 live tend to vastly overvalue their pocket pairs or think calling say $50 into an $100 pot is a "good price" to draw for a flush. Plus there's even competent players who thrive on mistakes of others giving them cheap draws to then exploit on the turn or river. Yeah, I get there's a specific strategy of check-calling to river when you then jam or overbet to take a pot, but what happens when you reach the river and your cards might be good but also might be now crushed? I just don't see much value in check-calling flops where we have a set , if they're raising me after I check, they're going to call my reasonably priced re-raise, why not build the pot?
Sure some players want to protect hands and shut it down when it gets scary, take the smaller pot and run. But think about your line here? It seems you wanted to let them catch something on turn, (maybe if they have AK and peel off an A on turn you get your wish) and bet into you , but why leave money on the table with a flopped set on a rainbow board? Check-raise the flop, if they come over the top you can decide if you want to go all in or fold. If they call your raise, continue the aggression, build the pot, and likely take it down.
Yeah in this specific hand, villain was playing you and trying to get max value. Somehow you got away from it, but most of the time when you flop a set on a board like this you're good to go. Might as well build a pot on each street and get as much as you can. In this sense, a free card on turn does nothing for you , if they're betting a draw or overpair or two pair on the flop, raise them and make them think you're trying to bully them out. Can't fear set over set.
