You know, maybe follow a program for a little while centered around olympic lifting. Have you ever seen an olympic lifter?
Yes many, Im one myself.
Those guys are truly amazing. I would think perhaps those kinds of lifts could translate to the bench, squat, and deadlift.
Yes but not in the way youre thinking.
They strengthen the posterior chain, upper back, shoulders. Let's say you bench, squat, and deadlift X amount. And you clean X amount because you've never really olympic lifted. But you increase 50lb on your clean. Do you think that would show once you go back to the big 3?
The thing with OL is that the line of pull for a clean or snatch is totally different than the setup and line of pull for a deadlift. Because of that, youre much better off doing speed deadlifts or something like that to get more power off the floor. Also, when youre DLing, you just need to grip and rip. When you OL, your pull is deliberately slow until you start accelerating at the knew, youre basically trying to jump with weight.
Do a jerk comes from your legs, not your arms. You need static strength to support the weight, but you can train that with OH lockouts if you need to. Even then, the lockout isnt where most jerks fail, its staying upright during the dip and drive.
Snatching MAY help a deadlift if your weakness is off the floor but thats about it. The percentage used on these lifts can be so low compared to your limit strength that its just not all that worth it. Again, speed work from the floor with that grip would help, but you dont need to snatch.
Yes OLers squat, and they squat alot, but we squat differently than PLers, using more quad than hamstring due to depth differences. Also, if youre ever going to use shirts, suits, etc. OL has even less advantage.
So basically, train at what you need to compete in. If you need to bench, then bench and focus your assistance around your weak points. But a completely different sport will not help you very much.
Last thing, the only way I can see OL helping is via the neural system. They can make you hyper-sensitive to firing your nervous system which helps when moving maximal weight, but only if you train OLing for a good length of time. This is why badass Olers like Vasil can do other things like SM or PL because his nervous system is really efficient. OLing is not the only way to develop this though, so take that with a grain of salt.