A couple of comments from what I've seen here thus far.
For the original idea suggested, I don't think it would cause any sort of balance issue to use what you have suggested. However, I do feel as though Sorcerer *should* have their own spell list. The problem is that the wizard list at present is just...better. Tweaking the sorcerer list to add a few spells, and perhaps knocking a few off of the wizard list would help fix this issue, but the problem is practically anyone you speak with will have a different opinion as to what spells should be on each list. I suggest you figure out which spells feel most "like a sorcerer" to you and go with it.
I do not actually think that the sorcerer is that bad off compared to the wizard. The problem is that there is a misconception based off of previous editions that the sorcerer has more spells per day than the wizard. Sure in older editions this was the case. However, the 5e sorcerer is NOT the same class anymore. The issue is that they standardized spellcasting by giving everyone the same number of slots and casting mechanics, while not giving sorcerers enough to compensate from previous editions. What the sorcerer needs is enough mechanics to constitute the idea of being a spellcasting who manipulates raw magical power. I do agree with the sentiments that I really wish they had combined sorcerer and warlock, or the idea that perhaps sorcerers should've been the ones to get arcane recovery instead of wizards, but we kind of have to work with what we have. Metamagic is a great concept. The problem we have is threefold:
1) Sorcerers really should have more than just two to use before level 10. To be honest I really think they should have just known the whole list of metamagic effects, or twice what they know now at least. That would mean needing to add something for their later levels to give them features, but that would be a great concept.
2) Sorcerers do not have enough metamagic points. All of you other suggested changes ultimately really amount to this one issue. Rather than tweaking with costs and other complicated things, you could simply look into ways to give the sorcerer more points. First, I really don't like the idea of sorcerer bloodline abilities costing points. They newer bloodlines seem to be doing away with this, but the issue is it just makes them more desirable than the old ones without any sort of update to the older ones. Second, the amount of points they give you each long rest is nowhere near enough to use them often enough, which is boring from a player perspective. An easy fix would make them function like ki, in that they are recovered after every short rest. However, that alone is way too powerful as the cost to using them as spell slots would likely need to be changed, as that is a fair amount of extra power if not (though this would make for a decent higher level ability, perhaps to replace the metamagic stuff that would get cut off in my above suggestion?). Another decent comprise would be perhaps to give sorcerers an "arcane recovery" type option, but make it recovering spell points (perhaps half your level, usually leave once per short rest?).
3) Wizards need to back the f*** off of metamagic. Like it or not, it's now the sorcerer's 'thing'. You have yours already, more spells known/prepared, more effective slots via arcane recovery, and ritual casting. This means no metamagic feats, and no wizard traditions with metamagic (looking at you UA Lore wizard with your Alchemical stuff!). If you wanted metamagic you should've rolled a sorcerer! >_<
Related note: if we are harking back to days of old, perhaps sorcerers should have some type of ability akin to the old Eschew Materials. Perhaps let them count themselves as an arcane focus for the purpose of spells?
Other related note: Seriously Wizards, back the f*** off! Why is every one of your new archetypes effectively stealing other class's thunder? I'm looking at you Thurge!