Standards for finite-field discrete log cryptography
So, why do the NIST/TLS standards not allow for cofactors greater than 2?


Actually, NIST does allow larger cofactors - see table 1 in SP800-56A - they allow the subgroup (q) to be significantly smaller than the modulus (p).

As for TLS, previous versions of TLS (1.2 and earlier) did allow the server to specify the group (and made no requirements about the cofactor).  In TLS 1.3, they allow only a handful of specific groups to be used, all with a cofactor of 2.  Is there a specific reason they should have included a group with a larger cofactor?