The big difference between pull and push is which system has keys to access the other, and what an attacker could do with them. With your home network you might ultimately decide this isn't too important, but it's worth at least thinking about anyway.

In a push setup, each machine has some way (likely an SSH key) to authenticate to the NAS and push backup files to it. Each server has a different key to access a different path on the NAS, so if a server is compromised the attacker only gets access to that part of the NAS data, and if the NAS gets compromised, the attacker can't connect to anything but has access to the encrypted backups (you do encrypt the backups you care about, right?). This limits how much extra data the attacker can read, but has the downside you mentioned.

In a pull setup, the NAS has to have a way to connect to each server, typically as root for file access permissions. This means that if a server is compromised the attacker doesn't gain a way to access even a limited portion of the NAS, but if the NAS is compromised they gain access to keys to root access on every server, which is likely catastrophic.

A compromise solution can work. Have each server back up to a local file, then give the NAS permission to retrieve only that file, rather than root access. Whilst rsync isn't going to work for creating the single file backup, something like borg or restic would. This does mean you need more disk space on each server, but it also means that the server doesn't need direct access to the NAS, and the NAS only needs unpriviledged access to each server, mitigating the risk of a compromise.