This strikes me as an attempt to balance characters. Sure you can have that Infinity + 1 sword crafted from the finest adamantite right off the bat, but then you can only afford armour woven from the less-than-finest sack cloth. In essence, doling out cash from the beginning as you describe is not terribly different to a skill point system designed to force players to specialise or become a Jack of All Stats.
I'm reminded of a school-based game I was once in a billion years ago, where every character had a stat total that had to be divvied up among exam results, extracurricular activities, and other completely external activities. What this basically meant was that every character would have strengths and weaknesses, which sounds fair, but, in my opinion, derails realism (which is something I tend to aim for); there are kids that breeze through school, getting great grades in everything and are the star striker of the football team and solo flautist in the orchestra. Is this because they work hard? Or maybe because they have psychotic helicopter parents and they're hugely stressed and are about to burn out? What about another kid who had a rough start in life, made all the right decisions and worked hard at school but still has a really mediocre statpool to show for it? How might they feel about the kid whose parents could afford private tuition? What might they have learned through their experience that the privileged kid didn't? The moment you apply arbitrary balancing mechanisms onto characters, I feel like a lot of this really human nuance is lost.
Clearly, nobody wants to play with an overpowered character, but 'more powerful than me' does not intrinsically mean overpowered. I suspect that when most games apply any kind of numbered values to characters, it's an attempt to stop people making OP characters and a disclaimer to stop people bitching that they feel that another character is overpowered. The thing is, where those stats are simply a reference, nothing stops anybody from playing overpowered anyway. If the GM wants to curtail overpowered characters, they need to communicate effectively from day one what constitutes a reasonable character and what constitutes an overpowered one, both via whatever sheets are being used and by indicating what the game is about and what sorts of things will work. Crucially, they should only admit players that "get" the game and whom they can trust to basically not abuse the privilege of being able to narrate their own story from their own perspective.
For me, there is very little correlation between video games and RPing. When a character in most fiction 'levels up', that's as much because of an emotional change rather than a physical change - the Eye of the Tiger training montage can only happen once a character has decided that they're actually really fucking determined / another character has agreed to teach them. The 'levelling up' is not just a device to make the character more powerful to progress the plot, but reveals something about the inner life of the character. By the same token, when a character acquires a new skill, that's frequently laying the seeds for a time for that to become relevant: when Harry Potter learns Expelliarmus and it becomes his signature spell, not only does it reveal something about his (mostly) non-violent inner life, it also becomes a plot point by Book Seven and comes loaded with a load of symbolism to boot. While, granted, some video games do do this*, the main reason for level-ups and new abilities is to maintain the difficulty curve and to keep the player's interest because they have some new shiny shit to throw at enemies.
Since, for me, RPing is a way of storytelling and not an exercise in the vicarious experience of fiction (I have actual video games for that), there's really very little to be gained from a graduated difficulty curve maintained by stat growths and gold acquisition in the face of increasing challenges. Particularly with tangible upgrades, such as a big shiny axe, keeping on top of what players have access to sounds like a fucking nightmare for a GM. How do you stop the players from picking up the Goblin King's Spear of Awesome once they've just killed him and his minions? Or, what, they pick it up, and it's actually a piece of shit? But the Goblin King was a tough enemy, so how does that square? What about the shop-keeper? I'm sure a ragtag bunch of rogues could overpower him and simply steal everything he has. There might be repercussions of some kind, but it sounds like a fantastic way to derail the plot.
It's worth saying I don't play tabletop games/NRPs, so this might just be a case of "the idea isn't my sort of thing because... it's not my sort of thing" - everything I've said above applies to non-tabletop/NRP games, where I'm sure there are really great DMs and GMs who can really run those sorts of games with whatever system works for them. I just know that arbitrary measurements of character aptitudes are a straightforward turnoff to any 'normal' game that I might otherwise be interested in. I'd be much more tempted to say "here's the setting; make a character; don't go mad" and see what players come up with. If it doesn't seem right, point out the bits that you're unsure about and see if you can reach a compromise. If you don't think they really "get" the game in question and they'll keep coming back with something that doesn't work, that means they're probably kryptonite to you and your game would be better off without them.
* I can think of a particularly good example where Rydia from FFIV refuses to cast the fire spell because of a pathological and justified fear of fire: only when she is forced to by circumstance, by which point she has come to trust the party, can she bring herself to even try. Then she gets access to Fire Magic in standard battles.
