A couple of thoughts:
1. On a normal drum kit, the overheads are the stereo recording of the entire kit. This is your main microphone. The close mics on the drums are just filling up what doesn't get across powerful enough on the main mic. You wouldn't consider what's picked up by the main stereo mic of a classical recording as 'bleed' just because there's so much space between those two timpany hits.
Following that logic, you will lose most of your recording if you cut or gate your main mic. It's not just a cymbal mic. So don't cut there. What you hear as 'bleed' is the actual drum kit in the room. Use it.
If you have more than one mic close to a sound source, there will be bleed. It's the nature of the game. You can reduce it by clever mic positioning and utilizing the mics' polar patterns, but not eliminate it, so embrace it, make all mics sound good *together*, bleed and all.
2. Noise Gates are among the most stupid volume controls there are, because they can not know which sound you want and which you don't want. Yeah, you can set the threshold well to choose between loud and soft, and use their key filters if you're lucky and good vs bad can be divided into hi/low. But once what you want is quieter than what you don't want, you've lost that battle.
Now, they may be useful on toms, if there are many and they are played often. But if there's only the occasional Pat Boone Debbie Boone leading into the chorus, then by all means, cut the recorded Region(s) and put long fades on their ends, just like a perfectly set up Noise Gate would, but this time it's you who decides where, how and what precisely gets cut and faded. Like this:
The time to actually edit this is less than it takes to plug in and set up a noise gate so it works correctly (which may even be impossible if you consider soft tom hits against loud snare hits), plus edits don't use any processing power, whereas a noise gate does.