Question: If I want to persist the users' login such that they do not have to re-login even after, say 1 year of inactivity, then is storing a permanent access token as good as storing the password directly (perhaps hashed), since the (permanent) access token would essentially be the "alternative password"?

Answer: 
Storing an access token is surely safer than storing the password directly, but let's see why:

1. An attacker can only get this token, but not the original password. This is better, because passwords are often reused on other sites, and/or can reveal password schemes. ➽ Make sure the token is random and not derrived from the password.
2. The token is not just another password. While passwords choosen by a user are often weak, a token is very strong. They are so strong, that brute-forcing is impractical. ➽ Generate random, long enough tokens, they should be at least 20 characters a-z,A-Z,0-9.
Is better to store an access token than storing the password or the hashed password (can always try brute force to find the password) and I think you should give a token a lifetime.