Are there any alternatives to a new router and a NAS?
Apple is pushing everything to iCloud services. Unfortunately Time Machine is still limited to local network or directly plugged in drives.
I think they will eventually get there.. so TM will no longer backup your Mac OS as that would just be wasted space in cloud, but all your user files and settings etc.
For now the best option is a USB drive plugged directly to the computer.. in your case a charging USB hub could be used instead of Apple charger and provide fixed USB drive for any occasion the laptop is charging.
As far as network backups go.. your choice is NAS for a number of reasons..
Routers including Apple Time Capsule do not support SMB3. That is the protocol default used by Apple since Sierra. Backups to Time Capsule still work over the AFP protocol and routers like the Synology or Asus etc that support TM backups. Only NAS have sufficient CPU and memory to handle SMB3 which is preferred for use with any of the newer OS.
The only alternative Apple made available for Network backups is using another Mac. The original Apple Server Extension for Time Machine was ported over to standard Mac OS since High Sierra. It is not so easy or cheap to implement unless you have a suitable Mac running Big Sur or later, and a large drive hanging off it via Thunderbolt or USB.
So a NAS is now the best solution, if you insist on Network backups.
Synology are recommended for best compatibility.
You can get a dual bay cheap one .. (single bay restrict you but will work) even second hand will be adequate to the task. Nor do you need to fill it. One big disk will be fine for backups. Only if you fill it up do you need to get another disk.
Note the format of the drive used on the NAS is not a problem. Or even USB formatted HFS+ plugged into Time Capsule or any other router or suitable format. Time Machine creates a Sparsebundle that is a virtual drive, and it will be formatted APFS.
The only setup where you can backup to APFS formatted drive over a network is using a second Mac as Time Machine target. And Time Machine will still create a Sparsebundle on the drive in APFS format. That is just how TM works.
The advantage of using HFS+ is the Mac can still read and write to it.. so if something goes wrong you can still plug the drive into the Mac to read the backup.