You don't really need your CA doing eIDAS in the system root. This scheme works as a closed system where you need eIDAS app to produce the artifact and another eIDAS app to verify it, when both have their own non-system root.

Ukraine for example successfully operates their own eIDAS-like scheme where everything is based on DSTU+GOST algos not supported by any operating systems a major libraries, the certs are signed by the government root and it doesn't leak into web pki.