I don't think I saw what your current weightlifting routine is in terms of sets and reps?

When I was younger and cared about lifting heavy weight, I was using a bench press routine I read in a magazine from some dude who benched like 700 lbs. It was 4x5 as heavy as you could, up to 5 minutes between sets,4x5 flat then 4x5 incline, followed by 4x20 flat dumbell flyes. That's Monday. Friday, easy 4x5 with about 50% of max weight.

A few years later when I didn't care as much about lifting heavy, I would do the "Bas Rutten" MMA workout M and W, and on Friday do something like 2 x 12-15 or more with as much as I could on bench. The Bas Rutten workout has tons of push ups in it, and doing all those pushups on M and W, then bench on Friday, I was surprised how heavy I was able to bench when I wasn't at all focused on benching. On good days I would do like 185 15 times, at a bodyweight of 145. I remember the one rep max equivalent was near 300 lbs, which was good enough for me. That's probably as much or more than I could theoretically max at back when I was doing the 4x5 with heavier weight.

I think I concluded at that point that I didn't really need any set routine, as what expert would ever recommend 2 days of pushups and 1 day of 2x12 in order to get a good bench press?

If you're not getting as strong as you'd like, either you're not lifting enough or you're lifting too much and not giving yourself a chance to recover. Nowadays I don't care about bench press #s so just do the Bas Rutten workout. One thing I have noticed is that if you have in mind how many reps you want to do, your mind and body will kind of work so that once you reach that point, your mind will tell you that you can't do any more, even though you can push past it if you're aware of what your mind is doing. So if I was benching still, I'd probably just do whatever I could safely on any benching day without having a number of reps in mind, and make sure I waited until a day after no soreness to go again, unless going very light if there was any soreness. Since I workout alone, I wouldn't push myself to the brink on bench for the risk of the weight falling. Which is why I stick to pushups now a days.