Since an SSD has no moving parts, it seems to me to be perfectly suitable as the root disk of a server that is always on.
Is there any reason that you would suggest not using an SSD for this purpose? Am I missing anything?
Short answer: no, I don't think you are missing something and using a SSD for that purpose would be a suitable choice.
Slightly longer answer: there are some concerns with SSDs as drives containing all the system areas because of the constant small block writing that goes on due to log updates and such. This really shouldn't be a problem with modern SSDs in terms of write counts and related failure rates though. It is a write pattern where SSDs can be relatively inefficient, due to blocks larger than your filesystem's blocks needing to be refreshed and rewritten potentially for each block write, but unless your log files grow at an extraordinary rate you don't need much throughput for these operations anyway so you'll be unlikely to notice this difference (and I doubt the effect would be enough to bring the SSDs performance below that of a spinning-disk-and-moving-heads drive anyway - though I have no scientificly measured benchmarks on the matter to link to or paste in right now).
The usual precautions apply if the server is at all critical: you might want to consider two drives in RAID1 for instance to allow for easy recovery should a drive develop a defect.
Most likely it the answer is that it would work, but many server vendors are scared/worried to change.
If you look around enough dedicated server hosting companies, they offer SSDs on the top of the line servers, but when it comes to servers in most (not all) environments, the weakest link usually is bandwidth not I/O, therefore putting in a SSD will just cost more money without having any of the other benefits.
Also, stability - they may work, but for mass adoption, it can take a while to "prove" itself - If you look at Intel Telecoms / mission critical, they only recently migrated away from Pentium 3s!