In general, there are lots of good thoughts in here. I don’t have an awful lot to add, but support for the concept. Anything that helps the 3-to-4 transition not take literally years is a plus in my book.
I agree with many of the great ideas and concepts above, but what i thought to propose is a bit simpler… possibly even a “that’s no skin off our backs, let’s do it” type of idea. (One can hope)
How about a ‘double-heart’ system, based on the modified heroic stores. In these special heroic stores there is one 4*, 3*, 2* each that is available at an increased x8, x5 and x2 odds respectively. When you get a token for these, even if you manage to pull a 3 or 4 star it is almost never the one with boosted odds, the increase in odds is that slight. But it is there! So why not let players choose the character they want at a slightly increased odd?
So pick just one character from each tier (and I’d limit it to characters that have less than 13 powers), but they’d have an increase in whatever store they are in. Whether it’s the heroic or elite or legendary stores etc, for that character(s) only, the odds would be higher. By how much? I’ll leave that up to the devs. This would be separate from the BH system of course, as such some characters might end up with 2 hearts. (Or it could be a little four-leaved clover for luck or whatever.) 
This might only be a relatively small solution, to the big problem, but it would be so easy to implement, and as the other draw rate odds stay the same I think it wouldn’t break the bank either, since people would still need to drop large chunks of hp or cp for significant results.
At any rate it would certainly help those in transition in helping cover the basic powers for their characters. And it certainly gives the feeling of ‘my needs are valued’ every time you do manage to pull your ‘double-heart’ character from the stack, regardless of the tier you stand on.
What do you guys think?
Year one for me, dazzler/emma release when I showed up. made it to (4) 4* champs. Black cat after 2 alliance event pushes, shes still fresh right now 
Chavez one at a time all year last week, carnage bought his cover pack during holiday sale first champed for months after prowler release, and fury is ready, but his vines not… at least 1on the vine…
But yes if I wasn’t stubbornly determined burnout/quit would have happened several times this year… got a lot of catching up to vets to do.