Why do you continue to multiply time constant by 5? This is irrelevant.

The 37% mark comes from the definition of time constant RC and is simply used as a starting point for the relay release point. The 50% mark might be a better choice. Using your graph, the 50% point occurs at 6V, 10sec. Hence the solenoid will release after 10 sec.

 

OK I must excuse when we deal with educational stuff here. However why not put the capacitor in the gate path?

You have some kind of layout or simulation software, so you can think about that.

Of course fictional circuits are used educationally, but...well...

 

We are not trying to improve on the circuit.
We are simply answering a question about circuit behaviour.

As you correctly suggest, there are many ways to improve this circuit.