Does the chain move freely on the bar by hand? If not, your rails could be pinched or your brake band is engaging when it should not be.

My guess is that you have something binding your chain if your clutch and brake band are getting hot. I am guessing that the clutch is slipping causing a heat build up your clutch and clutch drum. Your brake band should not be engaging the drum when the brake is off, i.e. the drum should spin freely.

Does your saw rev well? If not, check fuel-air system. Check that the filter is clean. Saws will be under powered and bog if they do not have enough air when throttled. Do a compression test if possible. Even better if you can do a pressure/vacuum test.

Clutches don't fail as often as other things, but when they do, the symptoms are more likely that your chain will run when you are at idle due to weak clutch springs. Or you might hear a pinging noise as the clutch arms intermittently strike the drum.

Others may have different ideas.

You could have some buildup under chain ( saw dust oil ) packs very tight under chain. Take piece broke hack saw blade clean out groove you tell when hacksaw blade sliding on metal. When bar wears are spreads chain pull down in bar bind this cause hot spot on bar you see where bar rails been over heated most time this around tip bar. Drag file down to much teeth getting to much bite bog saw and load clucth. If run saw with slack chain until jump off it can bent chain links and bind chain after reinstall. Putting chain on backward and trying run saw even if turn chain around right.

All of the above plus; is the oiler working right?

Maybe Chain brake band warped and not releasing the clutch is my guess. Had it happen to my saw.

First of all, he said that his chain is properly tightened, is sharp, and chain is getting properly oiled.
My question, is have you cleaned with compressed air, around the clutch, and the chain brake band area.
Next, when you hold the saw out of the cut, but in the cut, it bogs down. Sounds like you need a carburetor adjustment, or a rebuild kit put in it.
If it is adjustment, adjust the carburetor to pull under heavy loads. Have some one help you with this, and as the chain saw is running in the cut, turn the high speed needle, one way or the other. Once it is running evenly, while under load in a cut, you are set to go.
If it was the clutch, the chain wouldn't spin properly, because the clutch would slip.
Some new chains (if this is a new chain) will grab a quite a bit, once it is new, but a few chain sharpening will help with that.