I'm not an expert, and I haven't even slept in a Holiday Inn recently, but testing the basic operation is pretty easy, and I see no reason to spend $1500 for a water softener period.
Kinetico is on the web, with at least some information on the site, and perhaps some tech support like an owners manual. I didn't notice if the control valve, first picture I think, says that "it" is Kinetico anyway. If you can get a manual or tech support of course do what they say unless its call the local dealer and have them do it. You should be able to do whatever it is you do to set the time, and move time setting forward until the unit does a regeneration cycle.
Salt water from the brine tank should flow out the drain line until the brine tank is empty of water, then shortly after that the drain water should have all the salt rinsed out of the resin. My drain line goes in the laundry thing, so I can just stick in a finger and taste the water. When the drain water stops flowing, the brine tank should refill and make reasonable contact with the salt.
If all that is pretty much happening, the mechanics of the system regardless of age are still working. Could be even something dopey like regeneration time set wrong so that water use in the house is interrupting the cycle, and just setting the time correctly might fix it.
If the old control valve is messing up, I would replace it. From what I have read Fleck and Klack make very good units for $200 to $300 on the net. The thing in the brine tank is just a "version" of a toilet float and valve, and my guess is that its not expensive either.
Resin beds don't last forever, and some that go after stuff like iron I think are more fussy than the calcium units. If nothing shows up from the testing above, I would have samples of the before the softener and after the softener water tested. Then look into having the resin bed replaced, not the tank, just remove the resin beads inside and put in new ones.
No reason I can really think of that everything should be replaced, unless you just like the local plumber and want to see him happy. Control valve, brine tank, and resin bed can be treated as modules and replacing one doesn't mean the others need it too.
Finally just a bit on costs. I have been doing a little shopping, since I now have a culligan unit I have been paying $20/ month for the last 20 years on, and prices are all over the place, but unfortunately none quite low enough to tempt me. $500 installed would get me going, and for a quality unit I am seeing total costs installed of more like $600 to $800, with about $150 of that as labor. Of course its easy to pay twice that and get lower quality, some water guys should be flying pirate flags.
This is REALLY the sort of thing you don't want to do on impulse, ie open the phonebook and hand the smiling man a check. If your wife is ready to shoot you, and you need to act fast, I would just call Culligan and get a free install and pay month to month until you can think of a final solution (but I would still do all the checks above including having the water tested to see whats really in it you want out and how well the system is doing now). Here I am 20 years later though.