It’s always the case that low-value UTXOs may become immobile in a high-fee environment. Sufficiently small value UTXOs are dust—never worth moving even at 1 sat/vbyte. This is true no matter what enhancements we consider (base chain, Lightning, covenants etc.)

What covenants allow is more people holding keys per UTXO. So you could have relatively high-value UTXOs that are still movable in a high-fee environment, but which encode promises to pay a large number of key holders.

In the fullness of time, block space is unbounded. We’ve seen that even though occasionally fees become quite high, they don’t seem to stay that way forever.