I'm completely with you here, both tanks and supports are proof that some champions need to have different options during and after the laning phase. Riot has acknowledged this to a large degree with tanks, which is why many of them have been given ratios that allow them to build some damage early on, and then progress towards a proper tank build, or simply strong base damage with poor scaling. My issue with this so far, though, is that putting more damage on tank abilities tends to often make them too bursty, and even minor scalings tend to give them damage at a point where they really shouldn't be having much of it. This is less the case now, as top lane tanks aren't doing so hot, but back when tanks were really strong, champions like Poppy and even Nautilus were out-dueling damage champions at certain stages.
Personally, I think the idea of giving tanks and supports stronger damage early on, and less damage later, is a good one, as it would allow them to hold up at a time when CC and utility aren't at their strongest, without overloading their kits once they get to the teamfighting stage. However, I think the medium to achieve that should be basic attacks: since both tanks and supports tend to use basic attacks pretty often in lane, but very little afterwards, giving them solid basic attacks, and little to no damage on their abilities, would allow them to trade and last hit properly in lane, and naturally scale towards a state where their major contribution becomes their CC, utility, durability, etc. It could also make them healthier in lane overall, as limiting their damage threat to autoattacks could prevent some supports from using their more reliable CC/utility to harass enemies at too little cost (e.g. AP mid lane Janna way back when, Lulu before all of her nerfs, pre-rework mid lane Soraka, etc.).