I'm torn.

The way you describe it, some of the more casual players could definitely need some help and this would be a decent way to achieve that without creating inflation and throwing free stuff around like candy. It's a well thought out proposition that would give something to the people who need it without taking anything away from the people who don't, and it might be more relevant than ever in the FHM-era since the start can be tougher for lower TPE-tiers.

On the other hand I have to agree with Muted, being a high or max earner shouldn't be easy and I'm not sure we want to make it easier when so many people top out around 1500-2000 TPE already. Different tiers of activity and committment need to lead to different quality players, instead of making sure that there are no sub-1500 players anymore, we should look for ways to make the league enjoyable for them as well.

Also, if what you are saying is true about players hitting a wall after their first season, when training costs suddenly go up, wouldn't your proposal just move back that wall a few seasons while making it even higher? You rightfully point out that it gives players more time to build bonds and relationships that help keep them around in tougher times, but it also gives them plenty of time to get used to the more forgiving system and get hooked on the discounted training. Which will make the effects, the eventual withdrawal if you will, much worse when they are finally introduced to the real prices and will make it much harder for them to adjust because they have gotten so used to a different system. It is probably easier for a fresh player who is still in his first season or second season and excited about everything to get used to doing media or graphics when he needs some cash, than for a player who was coddled for four season and could get by without it so far.

So I am fairly critical of the idea right now but it certainly isn't without merit and the problem you describe is certainly legitimate. Maybe it would be worth to adress it from an income perspective instead of the expenses perspective? More contract money in the lower tiers, a slight rise to media pay either in general or for newer players. I could think of a couple of systems there, for example giving each new member three double media articles to be used on any piece he wants during his first 4 seasons or so. Maybe limit the eligible word-count to 5000 words or so so that it doesn't become expensive, but it would allow players to build a bit of a cushion early on while still requiring them to show their committment by doing some work. Or, as I said, just raise media pay in general, or be more generous with bonuses etc...

And regarding contracts, one idea that I have thought about a bit already that could apply to younger players, but might also work as a general concept for the entire league, is to detach cap-hits from actual salary to an extent. This would work through a simple multiplier, like x2 for example. That would mean that a player who counts with 2M against the cap actually gets 4M, a player with a 1.5M cap hit gets 3M etc. In the lower tiers it would give people some extra money but in the higher tiers it would probably be overkill, so we would probably need to have that multiplier only apply to lower tiers and either let the factor drop the higher we get in tiers or get rid of it altogether. It could look something like this

Tier 1M x 2 = 2M
Tier 2M x 2 = 4M
Tier 3M x 1.667 = 5M
Tier 4M x 1.5 = 6M
Tier 5M x 1.3 = 6.5M
Tier 6M x 1.25 = 7.5M

This would give people an extra boost early on when they need it without creating a wall that they hit at some point even as the multiplier drops. However keeping the multiplier consistent at x2 could also be something that spices up FA in higher tiers, after all, we want people to sometimes go out and chase bigger deals instead of always just taking team-friendly minimum contracts. If the difference between a 6M and a 9M contract would actually be 6M instead of 3M, that would probably help accomplish that. It would probably lead to a bit too much inflation though and throw too much money at the people who already have enough.

There also is a different version of this that is simpler but also more radical: Don't have a multiplier or anything, just raise the contract tiers to higher numbers in general and raise the cap accordingly. This would require quite a few calculations to find a new salary cap number that works and probably wouldn't be feasible in a system we just changed, but it would be the cleanest solution. I think having contract tiers be a bit more apart instead of being all in 1M steps could be pretty cool and add a bit more character to things if say tiers looked like:

2M
3M
4M
6M
9M
12M

The cap would have to be a good 20M higher or so than it is today. Superstar players would take up a bigger chunk of your cap than they do today, just as in real life, but overall all teams, even the good ones, would have a bit more money available and more flexibility to play around with and try different roster building approaches. Some might make a superstar heavy roster with some cheap filler players to complement them (kinda like Pittsburgh did with Crosby, Malkin and their linemates), while others go for a more balanced approach with strong players in every position at the cost of not being able to afford more than one or two 2000 TPE players.