By my perspective, the issue is less about auto-scaling cantrips, and more about auto-scaling utility spells.
At low levels, you can cast Burning Hands or Shield from your level one spell slot, and they both give a level-appropriate effect. At high levels, you still get a level-appropriate effect from casting Shield in a level one spell slot, but casting Burning Hands in that spell slot is basically a waste. If the benefit of casting Shield was as trivial as the effect of Burning Hands, then we could all just agree that level one spell slots are a meaningless resource to high-level casters, and ignore them; but Shield is still useful at high levels, which means we know what that spell slot is worth, and Burning Hands does not return that value.
Cantrips are mostly fine as they are (from a mechanical standpoint), but low-level damage spells are such a terrible option that they make cantrips look overpowered by comparison.
1) Should cantrips not scale any at all?
You could do it, but it would require re-working the HP system such that ~10 damage remained relevant at higher levels. It's probably more effort than it's worth.
You could also keep cantrips static, if you let spellcasters upgrade to level 1 or 2 spell effects at higher levels. If a level 5 wizard stopped casting Fire Bolt at-will and started casting Burning Hands at-will, then damage would scale appropriately, and you wouldn't have to worry about your at-wills overshadowing your limited resources.
2) Should spells that end up being strictly inferior to scaled cantrips scale themselves at some point?
There's no reason to limit it to lower-level spells. All damaging spells should auto-scale, in order to remain level-appropriate for their spell slot. If Fireball and Haste are both worth a level 3 spell slot when you're level 5, then you should get the same relative effect from each of them when you're level 13.
Of course, that would probably mean re-balancing spell damage to better keep it under what a fighter can do. You can't auto-scale Burning Hands up to Fireball level, since it would give you so many more spell slots to work with. Off the top of my head, I would say that all of your fire spells do 2d6 damage per maximum spell slot, but it would need more research to be sure. The basic idea is that a level 9 wizard could deal 10d6 damage to a single target at-will (per Fire Bolt), but 10d6 in a small cone with a level 1 spell slot (per Burning Hands), or 10d6 damage in a 20' burst with a level 3 spell slot (per Fireball).