The whole of early game needs to be addressed
There is such a disadvantage to picking any champion who doesn't excel in early game, that carries into mid-late game. Even if that champion scales better.
The speed at which you scale up in power is so fast, that being an early game champion is irrelevant, because you can enter mid-late game with such a disparity in power that even with worse scaling, early game champions keep ahead in power due to more items obtained from their implicit early advantage.
Completed items just create far too massive of a power disparity from the component parts. Finished items need to have their power toned down, or the component parts need to more frequently offer similar weaker versions of the same passive.
E.g.
has a a weaker version of 
unique
doesn't get a weaker version of 
unique
The disparity between raw stat numbers and unique passives has become so large that it's nearly impossible to argue that having 1/2 of the AP and all of the armor is anywhere close to as valuable as a miniature unique.
Then to top it off, by making tower deaths so frequent and early, along with the new dragon buffs and rift herald, you've got 3 different objectives in the early-mid game which can't be offset by the losing team. No amount of gold will give you +8% AP and AD, into the mid-late game. The tower situation is compounded, because jungle control is heavily based around tower conditions. So losing T1 mid and bot tower by 10 minutes, heavily stacks the probability of reliable dragons in favor of the team with more early power and aggression.
There's no easy solutions to these problems, because these problems weren't easily implemented. They were added over years of riot pushing more and more of them into the game. Then simply adjusting the numbers of champions offsetting the problem so that they reflect a "balanced" state, of a reasonable win rate.
It's going to take a lot of changes to fix and a lot will seem like riot changing things back, which is probably true, because a lot of these problems were immediate unstable solutions to pressures leading to the time of implementation. Bot side too favored? Add an objective in baron pit before 20min. Dragon boring? Give it these snazzy buffs which act as a multiplier to effective gold instead of a flat increase. Vision stopping ganks (and therefore tower pushing) between 1st and 2nd tier towers? Rework the ward system.
Edit:
To be clear, i don't think that this early game bias was necessarily intentional by riot, but you could make a case for that being true. I think it's just a natural side effect of a bias towards multiplicative power increases over flat increases. The reason being that am equivalent multiplier for the winning team is more valuable than for the losing team.
Say the team gold is 5k to 7k.
The team with 5k gets a dragon, that acts as a 20% gold multiplier for the team, effectively increasing their gold by 1000. Meaning the absolute change in power disparity is now 1k gold. Where as if the 7k gold team gets the dragon, it still acts as a 20% gold multiplier, effectively acting as 1400g. Meaning the absolute change in power disparity was 1.4k gold.
Taking objectives for teams ahead is simply worth more than taking objectives for teams behind, when it's multiplicative changes in power as opposed to a flat change.
Starting out from that same game state. Say that you still had a flat gold reward from dragon (i think it was like 1125 or something before the dragon changes), let's just say 1k.
If implemented in such a way, then if the losing team got the dragon, they'd see roughly a 20% power increase (if gold was perfectly reflective of power). Where as if the winning team got the dragon, they'd only see roughly a 14% power increase.
Now, this is easy to show mathematically for the dragon buff, but it exists for a lot of the systems in the game now. This is arguably the root cause of the game's snowballing, flat values are relatively more beneficial to the weaker of two teams/champions, where as multiplicative values are relatively more beneficial to the stronger of two teams/champions.