If the ligament is calcified, then it should be removed, but if it’s not, then is usually left it, but still has to be detached from the styloid process, as it’s attached & to shorten the SP it has to be cut anyway. It doesn’t seem to make any difference to swallowing, voice etc.
After surgery my voice was fine straightaway, no difference, & the same with swallowing. Chewing & opening my mouth wide was uncomfortable, for a week or two afterwards.
I think others have found that talking was a strain though, for a couple of weeks afterwards, we’ve had quite a few teachers who’ve found this hard so have had to delay going back to work.
Here’s a link that you might find helpful:
Hopefully others might chip in with more info!