Okay, next couple questions. First, is the photo below showing enough salt in the (softener)? Not sure what this part of the system is called.
Photo did not make it, but the place you put the salt is called the brine tank. If there is salt above the water level after the water is added during the brine fill cycle, that is enough. If things are working and you are not planning to clean the brine tank soon, you can go much higher.
Second, is it possible or worth it to install something in line to the kitchen faucet (and maybe other water outlets) to reduce the TDS or raise the alkalinity of the water?
A reverse osmosis filter will reduce the TDS a lot. Get one with a permeate pump for better performance. Some come with a remineralization filter stage, and that would raise the alkalinity.
Based on the responses above, my noticeable lack of water pressure (feels like 30psi) upstairs in the shower could be the filter needing a change?
Yes. A clogged filter would reduce the flow available for the upstairs shower. It would also reduce the flow available for the downstairs water use. If the upstairs loads are 10 ft higher than the downstairs loads, that would account for a 4.33 PSI difference. Going to try that pressure gauge? Have you put a tape measure to your filter sump (the big part that unscrews)?
On the cartridge filters that I am familiar with, it is possible to remove the cartridge, and put the sump back. So no significant filtering in that condition. You could measure the the cartridge and study what elements are available at leisure.
I might try to have that O-ring and silicone grease available before opening the filter and measuring the cartridge. If the person before you lubricated with petroleum jelly, the old O-ring would have swollen, and you would not get it back into place. I use silicone grease when closing my housings with 4.5 inch x 20 inch cartridges. I have opened a cartridge about 12 times and I have needed one O-ring when putting things back together.
Or do you guys think the pressure tank being so small is just not supportive of what's supposed to be a 2gpm max shower head?
No. It is the pump that determines how much water is available for the shower flow. It is the pressure tank that determines how often the pump must run.