Should I & how do I use a mash tun brew kettle with a false bottom?

The short answer is Yes. The long answer is... I have a 15 gallon polarware kettle w/false bottom and a 10 gallon of the same. I've found that I need the 15g for both mashing and boiling of 10 gallon batches-- need the extra headspace for the mash and potential boil overs in the boil. So I mash in the 15, then sparge into the 10 with an extra stainless pot for overflow. Then I hike my 15 up to the compost heap and empty the grains. I clean the thing and then pour my wort from the 10 back into the 15 and boil it up.

YES, there's some hot side aeration happening here. But my beers have been working out very well. It's a pain in the butt though. If I had 2 15gal kettles, I'd be a happy boy. But I don't at the moment... and this system works, though it's time-and labor intensive. Especially since I'm hiking up a very steep hill to the compost :)

Seems reasonable. I usually clean out my mash tun while I'm waiting for the wort to boil, so you would lose some time. All-grain brewing is already more time-intensive than extract.

Instead I suggest you do a little research into the brew-in-a-bag technique. The method is nearly the same as your combo-tun-kettle, except the grain is in an easy-to-remove giant sack. I've used it a few times for 2-3 gallon batches, but you can do full 5 or 10 gallon batches using a bag.