With Avis (and presumably Hertz) you should be able to go straight to your car. I rented with National once in 2018 or 2019 and picked a car off the aisle and was on my way pretty quickly. Hertz would be the quickest exit to the highway, followed by National and then Avis, but the difference is 1-2 minutes at most.

I know you said you won't do EWR, but by going to JFK you are easily going to pay $10+ in tolls. Between bridges, tunnels, and congestion tolling in the city, it is going to get very expensive very quickly. Personally I think you are making a mistake going to JFK. Even if you go to HPN it is one bridge to cross the river and then you're home free.

-J.