Covenants are not a “solution” to high on-chain fees. I would not make that claim. If others are making that claim, I don’t know their rationale.

Covenants are a means of increasing the number of people who can hold keys on chain. If anything, this is likely to make fees higher on a per-UTXO basis. But hopefully lower on a per-person basis, as fewer transactions actually settle to chain.

For example, in Lightning, to update channel balances, peers share transactions that are valid but private. They aren’t broadcasted. You don’t open a LN channel just to do one transaction. Rather, you open a channel, do many transactions, then close.

By allowing people to share UTXOs in covenants, similar update schemes are possible. You wouldn’t just join a shared UTXO to immediately exit. That would be worse than just using a regular on-chain UTXO, just like it would be worse in Lightning to open a channel, do one TX and close.

So yes, in your example, the exit may be costly, especially in a high fee environment. But hopefully exit will not need to be frequent. If exits are frequent, then there’s little to no value in sharing (it may even be worse depending on the scenario).