Hot water heater running out at 50%
You absolutely cannot delay delivering cold water into the tank to replace the hot water output. You have a pressurized tank which must remain full of water to deliver water. If you had an unpressurized water tank which delivered hot water by gravity, you could do this, but you don't have that.

I assume that the cold feed is very cold in Cambridge MA. You may not be starting with a full tank of 130 F water. It could be that the controls on the boiler are set to reduce the number of heating cycles per unit time. Where is the temperature sensor located vertically in the tank?

Some recommendations on the temperature set point of tank water heaters are that it be set at 140 F (60 C) to prevent the growth of legionella bacteria. Other recommendations give 130 F. Does your shower valve have an anti-scald  temperature control?