What would cause set_transient() and set_site_transient() to fail silently?

If you have a persistent object cache setup, then transients will not write to the database: they’ll be stored in memory. Reviewing the definition of set_transient() this can be seen:

if ( wp_using_ext_object_cache() || wp_installing() ) {
    $result = wp_cache_set( $transient, $value, 'transient', $expiration );
}

Only if no external cache is in place do transients write to the database.

Try setting a value with set_transient(), and then attempt to recall the value using get_transient(). If you receive the value back, then there’s some sort of cache running. If you don’t receive the correct value back, then there’s something else going on, and will need to dig into set_transient() to see what’s going wrong where.