Pressure tank rusts, leak, and draw down questions
Visually the system looks dated


The tank itself is glass lined you can test this by using a magnet. Lower grades of stainless steel are still magnetic but only slightly, due to them being only moderately ferrite. Glass lined tanks are only good for 10 to 15 years if maintained well they can see 20 but it's very uncommon. By the look of the tank there's no anode in it therefore it would only last 10 years max
Glass-lined tanks can and will indeed rust and they will Rust completely through if given enough time. Low grade stainless is capable of rusting but usually to a lesser degree.
Losing pressure without a leak is most likely due to the water going back down the pump. It's possible that there is a leak in the tank at the top weld allowing a small amount of air to escape without being evident because water hasn't made it there yet the possibilities are endless.
Bladder tanks such as the Aquaflow mentioned are far superior to standard expansion tanks. The downside is perforations are possible rendering your expansion tank relatively useless but with proper maintenance perforations are very uncommon.