Getting a better contract, probably not big at all. That's more about the company being more verified than someone just signing up a landing page. Negotiating a better rate? I really don't know. We did it at ~6 figures per year in the very early stages of Stripe, but we were a UK beta account for them. I believe we continued to renegotiate at 7 and 8 figures per year, but we already had that relationship.

By all means use another payment provider as leverage to negotiate better contracts and rates, but I think it's unnecessary from a business reliability standpoint until you're very big and Stripe's downtime level (very small) is a material business risk. Spoiler: almost no one is this big.