Hi, I was a Panasonic dealer for many years.
The KX-TD816 is a very old system, but- it is very robust and good as long as you don't need any new features.

Chances are your fax does not run through the phone system, in which case your fax issues would have nothing to do with the phone system.
If, by some chance, the fax was connected through the system, it would have been done in order to save on phone lines for instance, but since you say faxing is critical to your company, it probably was not set up that way. If you need to dial a 9 on the fax machine prior to entering the fax destination number, then yes, it might be run through the system.

Feel free to send questions to me directly if you wish. Even many current Panasonic dealers may not be familiar with the old 816 system (the TD816 and 1232 systems were introduced around 1996).

As far as replacements, the TVP voice processing system needs to be replaced asap. The hard drives used in those system had an expected life span of about 5 years. The TVP systems were replaced by TVS systems probably 25 years ago.
The TVP will keep working until it doesn't. At that time you will lose all voicemail messages and be forced to upgrade anyway. Not trying to scare you into new purchases, but better to prepare than to be taken by surprise.

The 816 will also keep working fine, until one day when it doesn't. You will not find any techs who still work on those systems. The phone models are available on ebay, so you won't have any problems there, but a typical problem would be if a lightning storm knocks out some of the circuit cards or if the power supply goes bad one day. They are not hard to replace, but it will be hard to find someone with the knowledge or willingness to work on it.

Feel free to ask more questions here or contact me by PM
Charles