Half right. While 6-7mm is probably not enough, you don't need an inch either. 1/2" free height is plenty, and you can get by with less in most cases.

All the action of a seat post collar happens at the back, so all you is enough for the clamp to wrap around the front and stay put. Since clamps are typically 1/2" wide, then all you need is that much cylindrical extension above the weld or braze at the front. You have to be sure that the clamp fits low enough that the tube comes to or very close to the top of the clamp, otherwise it can cam and work upward when tightened.

There is no need to extend the slot, but you do have to have the right ID collar to match the seat tube, and the back area where the ears were welded has to be filed down to the round contour of the tube. If you have a lugged frame it's more complicated because it'll be a non-standard outside diameter, and it'll take more work to establish the cylindrical shape needed. and work upward when tightened.

In a borderline situation, like only 8-10mm workable height, it might be possible to carefully file away from the front of the clamp to create a saddle that fits over the weld, fillet or curve of the top tube. Obviously this will weaken the clamp, but it's at the area of least stress so there's some wiggle room to work with.

If you do file away significantly from the front, you can lessen the chances of the clamp breaking by gluing it to the tube. This will transfer some of the tension to the seat tube easing the stress at the front.