Meanwhile, the Rogue is gaining an extra d6 of sneak attack every other level, with no limit on the number of uses per rest, no spell slots to make it happen. Some small situational arrangement must be made, but those aren't hard to get.
By comparison a single d6, with no scaling with level, a limited number of times per day, falls behind really, really fast.
My response to that would be...
This is the inherent (and oft overlooked) advantage to a class that doesn't rely on magical power... This is exactly the "edge," forgive the pun, of a mundane (vs. "magic dependent," not meaning "boring/run-of-the-mill") class.
When a paladin is "out" of smites per day (or whatever increment) they have spell slots to fill in for some time -and to
@Stormonu 's original desire- and still would have some ability to use non-smite spells.
When THOSE are all done, a paladin is still a warrior archetype, right? Probably in heavy armor. Probably with a big weapon. Probably with a BUNCH of hit points (with the great AC = stay standing longer = more rounds to deal damage), and decent strength bonus. A paladin without smites is not somehow "woe is I, I can't deal decent damage, at will, all day long."
The thief with the sneak attack isn't laying on hands or have divine health or access to spells (as a default, arcane tricksters hardly count as they're third casters and paladins are half) or sense Evil ("divine," whatever) or project Auras of anything.
I was also, and probably should have been more clear,
not saying it HAD to be d6. It could be a d8. Or 2d6. Or "2 (automatically) + d6" (so, guaranteed, 3-8 hp extra damage). But, point was, whatever it is decided to be, it is flat and doesn't scale. If you want it to scale, then that's what the sl=pell slots are for. 1st level slot = additional [d6/d8/2d6/whatever it is]. 2nd level slot = additional [d6/d8/2d6/whatever it is] X 2. and so on. That's where your scaling is/comes in. So, it's possible.
But again, it's possible as a MAGICAL extra thing. So, no, it should not "scale" like a sneak attack. It's a limited -very special, even extra special if you want to take narrative/in-game from your [oath/god/ideal/whatever] source into account- resource.