The incentives are perhaps better aligned with a new token, but the market cap of ZAX would likely be much lower than that of ZEC, and I think the economic security would likely be lower as well. I think there would be a lot of people holding ZEC willing to bridge it over to stake, but who would not be inclined to do so if they had to buy another token. Ultimately, I think it is better for the Zcash ecosystem to have ZEC.z as the staking and gas token.
There already was a guy who ported the Zcash protocol over to an Avalanche subnet. RFI – Zapa - Community Grants / RFI - RFP Grants - Zcash Community Forum. Apparently it was working on testnet, but has since been abandoned. I don’t think combining this with a bridge to Zcash is an unreasonably large project. Maybe you can even collab with the guy?