The NFL isn’t going to say “don’t show other leagues”, that would cut their own throats. Especially since the arena league is already on Fox Sports regional and NBCSports. Throwing their weight around trying to limit what the channels show outside of NFL season would alienate their biggest paying customers.
Yes you can be non-regional as a D league. Remember the NBA D league is actually OWNED by the NBA. It’s very different from the other D/ minor leagues where teams might be owned by major league teams but the leagues themselves are independent to varying degrees. And subsequently can negotiate independent TV contracts.
That's exactly what they did to the first USFL and what they did to the WFL. ESPN did not have the NFL when they signed the USFL and the WFL was covered (poorly) by TVS.
The NFL goes to their tv partners and urges them not to carry competitors or risk possible losses in carrying future Super Bowls, etc. That's why, if the new USFL is truly a D league, they would do better to partner and be covered by the NFL than they would be to run independently. If the intent of the new USFL is to be an independent "spring league", they will need a strong tv network coverage and a presence in some major media markets.