The QAT_RSA implementation is uses OpenSSL's internal data structures, breaks with OpenSSL 3.6

The QAT provider breaks in OpenSSL 3.6. In openssl/openssl@902568b ("Remove need for BN_BLINDING lock") the layout of an internal data structure has been changed and this data structure has been absorbed by the project. This is not going to work as it relies on the private definition.
You need to either use the anonymous RSA type or use your own and if you intend to use OpenSSL's data structure you need to convert it back and forth EVP_PKEY_todata() and EVP_PKEY_fromdata(). See https://github.com/openssl/openssl/issues/30473 for the report on OpenSSL's side.

Sebastian