I have a feeling I'm missing something obvious, but the Wild Magic sorcerer's Tides of Chaos feature (PHB, p. 103) states:

Starting at 1st level, you can manipulate the forces of chance and chaos to gain advantage on one attack roll, ability check, or saving throw. Once you do so, you must finish a long rest before you can use this feature again.

Which suggests that this is a once per long rest ability.

But it goes on to say:

Any time before you regain the use of this feature, the DM can have you roll on the Wild Magic Surge table immediately after you cast a sorcerer spell of 1st level or higher. You then regain the use of this feature.

Which seems to suggest that you absolutely can use this more than once per long rest...as long as the DM has you roll on the Surge table after casting a spell of 1st level or higher.

Does that make it so that this feature is only once per long rest if the DM (or general wild magic) doesn't have you roll on the surge table before the end of the long rest? Otherwise, it's as many times as you want as long as you recharge it with a surge roll. Is my understanding correct?

Yes, it bypasses the limitation

Tides of Chaos states that it recovers on a long rest so that you always have it available to you at the start of the day regardless of its availability prior to the long rest (i.e. you might have been completely out of spells).

From playing a Wild Magic Sorcerer, I've found that the feature is an inherent part of my play style, but to use it I am dependent upon the DM permitting the frequent surges. In general, I strived to use Tides of Chaos almost every round so that I could utilize the recharge when I cast a spell. This is sensible, as a lot of optimization strategies for Wild Magic revolve around being away from your party and close to the enemies for variable shenanigans (which would be a good name for a band).

As has been previously stated, Wild Magic Sorcerer requires cooperation from the DM to be playable and enjoyable; if your DM doesn't permit you to recharge Tides of Chaos regularly with surges, you often are a lot more ineffective than other origins. If your DM is forgetting about situations to trigger surges, it's prudent to remind them; if they are hardly ever permitting surges, I strongly recommend playing a different origin as you will struggle to be effective.

Yes, Tides of Chaos is 1/day, plus however many times it gets recharged

Or, your understanding is correct. You regain the use of the feature after a long rest, and you also can regain the use of the feature by the DM giving it to you (by making you roll a surge).

Your first rules quotation describes the general use-case for the feature (use it 1/day), while your second provides an exception (additionally use it whenever the DM wants to recharge it for you).

Tides of Chaos' usage frequency depends on how your DM lets you use it

The Wild Magic specialization is designed to be erratic, and Tides of Chaos adheres to this. If the DM does nothing, yes, the ability is usable once per long rest.

However, if the ability is "cooling down" (i.e. has been used previously and you have not yet long rested to reset it), the DM can give the ability back to you without you having to long rest. But as a requirement for this feature, you must roll from the Wild Magic Table.

So the amount of usage tour will get out of the ability depends on how often the DM decides s/he wants the sorcerer to have access to it.

In one extreme, the sorcerer would be able to use Tides of Chaos only once per long rest. In the other (with a very permissive DM), you could use wild magic every turn of combat and roll on the Wild Magic Surge table to reset it for the next turn.

From personal experience, it is best to work out a system with the DM before the campaign so you have an idea of this.

Based on my experience DMing a table with a wild magic sorcerer, a cycle of advantage --> spell --> surge, repeated multiple times per combat, limited only by spell slots, is too powerful at lower levels. Advantage is a powerful bonus, and some wild magic surge effects are very powerful too.

For this reason, I take "you must finish a long rest before you can use this feature again" as gospel. Under this interpretation, the long rest must occur, but in addition, the DM may require a roll on the wild magic surge table before Tides of Chaos can be used again. I always have my sorcerer roll for a surge on the next non-cantrip spell she casts after her long rest. She can then use Tides of Chaos again. This gives the sorcerer one free roll with advantage, and one guaranteed surge, per long rest, plus a 5% chance of a surge when other spell slots are used, as opposed to many rolls with advantage and wild magic surges. This balances the wild magic sorcerer with the other characters at my table.