I think we are losing track with the point buy vs leveling focus.
No, I'm not losing track. I'm putting that focus there to make something clear: You don't want to reinvent the wheel, you do want to learn from history.
The point is that balancing a game based on small individual bits is difficult. The point-buy people have been at it for decades, and the systems are still prone to abuse.
At this point, it is hard to find a game design reason for leveling to exist.
I've already stated a game design reason - it helps regulate the power of characters. Levels take away the work of trying to balance a large number of small bits, by aggregating them in predictable ways. If you have a party of characters all of level 5, you have a significant amount of information about what they can handle. It helps players create characters that are not too ineffectual, or too potent in particular areas, rounded for the gamut of things you are apt to see in a campaign.
Character powers rise linearly, but the game assumes that the characters face linearly more powerful obstacles or problems.
(I think you're abusing the word "linear" here. But I digress...)
This simply isn't true. In 3e, 3.5e, and 4e, the rule books suggest that while the encounters should scale in a broad sense, that individually they should vary from easy to difficult. In 3e, the book specifically said that about 15% of encounters should be ones that the characters should run away from. The levels help the DM figure out what kind of encounter is one that the players can and cannot handle.
Furthermore, the "sandbox" or "status quo" DM typically does not scale encounters to character level at all. The party faces whatever they find, and if it is too easy to be interesting, or too hard for them to beat, well, then that's what happens.
I point this out not to say there's no value in your ideas, but to note that there's some false premises, to prevent you from working to fix things that aren't actually broken.
Any number of GMs could perhaps use some advice on how to manage character growth relating to things that aren't combat-focused. Tearing apart the whole system and starting from scratch with entire new concepts of development to achieve that seems to me rather like knocking down a house because you want to put in one room. Sure, you can do that, but it perhaps isn't the most efficient approach.
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