would still hash to the same fixed length even if the password were 1000 characters long
That's a big assumption. At one point at work we had a system (thankfully dead now) for accessing customer data that limited passwords to a max of 8 characters, only supported three non-alphanumeric characters in those passwords, and shipped a .net dll with every desktop client install that had both the password encryption and decryption function in it and accessible to anyone who loaded the dll. If you had user table access and that dll, you owned the whole system.
Fucking clown show.
Then they finally allowed longer passwords on the software side, but it was causing problems on the DB side. Because different length passwords hashed to different length hashes. Not one to one, but something like 1-4 in 14 out, 4-12 in 23 out, 12-15 in 36 out. Some sort of pattern like that. That was fun to troubleshoot.
Point is, you can't guarantee hash length in unaffected by password length.