While pushups do appear to be very much like bench press and should have a carryover, I just don’t understand it and they haven’t worked for me yet. Seriously. I tried a few different ideas for programming pushups in on chest day and only found that I either overworked myself or got bigger but made no strength gains. But, there’s always a chance that I was doing the programming incorrectly(I do that occasionally)

Here is a resource that goes over ways to increase your bench using pushups:

https://www.livestrong.com/article/458560-do-pushups-increase-your-bench-press/