My advice would be to test your recovery on the machines you intend to use if the Mac mini fails
Regardless of any advice you receive here or else were
In answer to you're question yes it shiould work providing the hardware is similar
test your procedures for getting as close as you can to the working status the mini was before it's death
Better to discover any holes now Rather than 2 in the morning in panic mode "cause the mini has died
And your "rolling backup" restore procedure is not working
Make sure the machine(s) that are viable candidates don't end up with some other mission critical software on them
That you have to run along side your server
In no particular order...
Export your boot disk to an external disk using Disk Utility booted from an installation DVD, and test-boot from that. (Hold the option key during a power-up to get the chooser, and select the external disk.) That's (usually) a viable recovery path, and worth testing.
Your boot drive will likely run from a similar-enough Mac Mini or Mac Mini Server, but there are a variety of Mac Mini Server boxes that are too new to boot and run OS X Server 10.6.8. Test any replacement.
One of the common approaches for a reinstallation-style upgrade (if this hasn't already tried) is a clean install followed by a migration in from your existing disk; either directly (via Target Disk Mode) or from an external copy created via Disk Utility (as mentioned earlier).
If your disk isn't upgrading successfully, then there is clearly already something a little wonky. Or corrupt. And if Apple couldn't sort it out short of an installation, you're potentially operating on borrowed time; these wonkies can have a wonderful habit of (re)appearing or mutating at the most inopportune moments, too.
One alternative option involves cloud hosting or such, depending on exactly what you're using this server for; outsourcing the server and the hardware. (This assumes various details of course, and might or might not be feasible in your particular case.) Or yes, get the budget in place for a replacement (new or used) Mac Mini Server, and test with the migration path "mentioned" earlier.