Is there such a thing as training for power?
Does it have to be done with olympic lifts?
Can it be done with basic barbell exercises like squat press bench deadlift?
Can it be done with sets of 3 @ 7 (fast bar speed)?
First you probably need to define what you mean by power. If you mean moving the bar faster, that comes with more strength. In any case I’d think that the way to go about this would be to focus on getting stronger through whatever method you want, then focus on pushing through movements quickly.
Power is specific. So what exact movement patterns do you want to improve power in and why? What are your specific goals?
Assuming you want to improve performance in things like the standing vertical jump, building muscle and improving your strength is one of the few things that can make a big, lasting difference, the other being training the movement itself, and possibly depth jumps (which only really help with movements that use a stretch reflex, i.e. not a block start in sprints, and are inherently stressful, so generally only used in peaking for performance).
In the end, force production is all that matters - and generally someone has bigger and stronger muscles (if bodyweight stays similar) will be able to produce higher forces and thus jump higher etc., than if they were weaker.
Your possibilities for improving explosiveness can be separated into:
Strength in lower speeds - squats, hypetrophy.
Strength in higher speeds - olympic lifts, weighted jumps.
Stretch reflex efficiency - plyometrics - like depth jumps.
Efficiency in the movement itself.
Obviously this is an arbitrary separation of important qualities (they overlap continuously), but for the sake of the conversation I think it’s alright. Now training the movement works the fastest, but assuming the movement isn’t technically difficult (vertical jump), after a few weeks or at most months, there isn’t much room for improvement. Plyometrics are the same (and on top of that they are stressful). You’re left with strength training and possibly fast lifts like olympic lifts/weighted jumps. Both of those are useful, but there is maybe less room for improvement in the fast lifts compared to strength - assuming you have good technique and there’s little room for improvement in technique/getting under the bar etc (which can take quite long, but hopefully not too long if you do one of the simpler variations like power cleans), I’m not sure for just how long you can expect to improve in the lift by only doing the lift. So you’re mostly left with strength training, which is the one quality you can improve for quite a few years.
For the purpose of speed-strength, olympic lifts aren’t necessary - the purpose of the movement is to not be a completely weightless movement like a normal bodyweight jump, and also not a heavy squat - instead it should be something inbetween. Olympic lifts fit this description, but so do some other lifts.