It is possible that you have a fault on the ignition switch, or the wire to
the solenoid from the switch.
I have twice cured the problem you describe by adding a relay to this
circuit, so that the current required by the solenoid doesn't have to go
through the switch. Any cheap automotive 30 Amp relay will do, cost me about
£4.00.
You can prove this is the problem (if you can get to the terminals on the
solenoid) by bridging from the heavy battery terminal to the small spade one
from the switch. If this activates the solenoid reliably, then you do indeed
have a wiring or switch problem, basically a high resistance joint
somewhere, which the relay will cure.