Rationale for verification with cofactor $2^c$ in EdDSA?
It's flawed to imagine that &quot;more stringent&quot; is somehow stronger.  Actually ed25519-dalek's verify_strict method makes this mistake.
There are many plausible verification methods, which all define valid signature schemes based upon the same ed25519 keypairs.  You'll find them described in https://eprint.iacr.org/2020/1244 and https://hdevalence.ca/blog/2020-10-04-its-25519am
Almost none of those schemes agree perfectly with batched ed25519 signature verification however because batch verification randomly cancels out some cofactor tweaks.
If you employ ed25519 in a consensus protocol, like for accounts on a blockchain, then you'll want batched ed25519 verification that agrees perfectly with single ed25519 verification.  In such cases, you should use ZCash's ed25519 crate ed25519-zebra.  Afaik, all other ed25519 implementations exhibit variations between batched and single verification (as of march 2021).
I doubt TLS would adopt cofactor clearing verification, in part because cofactor-less verification runs slightly faster than cofactor clearing verification.  In fact, TLS could already batch verify ed25519 because one never has third parties check TLS handshakes in consensus protocols.  I suppose a CA could issue a certificate that worked only intermittently and only on some clients.