If a screen is available, the Mac mini route is great.
If not, you could contemplate a refurb MBA or perhaps a used but well kept M1/2 MBA. All you need is a port for attaching an external drive or a hub for multiple external drives. Since this is for backup of a limited number of clients and over wifi, 10 Gbps USB-C is likely plenty for the drive(s). Such a system should not run you much money at all and thanks to Apple silicon it should have more than enough oomph while still running quiet and reliably for many more years.
In fact, if Apple does indeed release a ~$600 A18-based MacBook as rumors keep mentioning, such a system could also take over duty as a “modern Time Capsule”.