Search for a recent thread titled "Death by hi-hat".
It covers most of what you are having trouble with here. These problems are quite common. Mic placement can help, but 99.9% of the time, it is the player themselves.
Next time you are tracking drums, think of the cymbals as made of glass, and if you hit them too hard, they will break. At the same time, while doing this, think of the snare, kick and toms as the enemy (just kidding), and you are trying to beat the life out of them.
This is an extreme way of putting it, but I bet you on playback, your issues will be gone, if not a hell of a lot better.
In other words, hit the drums HARD and consistently, and hit the cymbals much lighter. Doesn't apply to every genre of music, but if rock, this typically will work. Or it always has for me. And trust me, there will still be enough cymbal to work with.
No mic technique can fix a player's balance issues, they can help a little, but in the end, the drummer himself must realize that if everything is hit the same volume, the cymbals will be picked up louder than the back beat, or rest of the kit. If the player also then rides on the cymbals, this creates a constant sustained "din" of high frequency racket that drums simply cannot cut through cleanly.
It masks the attack portion of the drum sound, and robs your drum sound of punch and weight. There has to be peaks and valleys in the drum sound for it to be "punchy", and if a wash of cymbal noise is filling those "valleys" up, there is less apparent loud to soft dynamic going on, and the drums will sound flat.
You also will have a difficult time with separation, and not be able to turn up/down parts of the kit without turning up or down that cymbal racket.
I know the cymbal is a musical instrument, but when it is hit too hard or played in this manner, it can almost have the effect of mixing white noise into the rest of your mix.
In short, take a long hard look at how you or your drummer are playing. Everything starts there.
Just my opinion. Very common problem, and I see it all the time. The above approach has always worked for me.