RSA engine with inaccessible private keys - susceptible to any attacks?
Would this setup be susceptible to any oracle-kind attack which would allow extraction of the on-chip key $x$?


Not as far as we know.

Any such Oracle attack that could recover the private key would demonstrate that the "RSA problem" and the "factorization problem" are equivalent.  Currently, that is an open question; such an attack would be a noteworthy advance...