Hi hats cut through. No need for a separate mic. I like the less mic way too. One overhead, one kick and one snare mic is all I use. The rest is just extra work. I don't like mics too close to my drums, but I run everything wide open. If you muffle your toms a lot, you may need separate tom mics. But with wide open toms, the overheads make them sound great.
For a basic capture, you could get away with just 1 overhead and a kick mic. But a separate snare mic allows for some real mixing later. Overheads don't do kicks well, you have to have a kick mic. But overheads get the rest of the kit only as good as your own inner kit dynamics go. You have to mix yourself as you play to a large extent, because you can't just say bring the tom volume up later. You could do a little with EQ but it's not like you have a tom fader. It's easier to get a clean sound with less mics, but it limits how much fine tweaking you can do later. If you need all kinds of options for mixing down later then yea micing everything is the way to go. You could share a mic for both racks if you have them. So a mic each for kick, snare, hi hat, overhead(s) and 2 tom mics. 7 mics tops. One overhead would do for no loss of flexibility for a total of 6 mics. The less mics the less issues.