Plumbing – put the pressure tank over 200’ from the well head

At 2.5 GPM, your 2-3 minutes of shower time between cycles is using between 5 and 7.5 gallons. If your tank starts at 60psi and 7.5 gallons is enough to bring it down to 40psi, that means you're starting out with 15 gallons of air in your 40 gallon tank, and ending the 3-minute cycle with 22.5 gallons of air. I calculated this using Boyle's law, or P1 * V1 = P2 * V2. And of course these numbers would be different if your shower head is higher-flow and your 3 minutes of showering used more than 7.5 gallons.

I believe to get the most output from your tank, you want it to be nearly empty of water when the pump is set to come on. That means draining and charging it to just below the pump setting, as you did. If you wanted to get a little more cycle time you could probably even go as high as 38 psi. Note that with a bladderless tank, you're going to need to recharge periodically as the air in your tank is gradually absorbed by the water.

For what it's worth I think 2-3 minutes is well within your pump's design cycle time. 15-20 seconds was way too short and by noticing it and fixing it you've probably saved yourself an expensive pump replacement.

I realized as I was reading through the B&G design guide above what I think the answer is. The 12 PSI is necessary for a two story building on a closed loop hydronic system. Most household boilers are limited to 30 psi before the pressure relief valve will open. So if that system were sized for the 60ish psi, you can probably imagine that there would be some issues. An expansion tank for a hot water system, on the other hand, WOULD be charged to the line pressure since it is directly connected to the incoming cold water line.