"Yes, motor fires and will run endlessly if I keep the ignition on start especially when giving a bit of gas but when letting go of the key it dies"
One each new starter and solenoid please.
Holy Cow, did I just read what I think I read? The start position engages the solenoid and sends the pinion gear into the flywheel. Are you saying you ran the motor with solenoid engaged? Never heard of that actually. Don't do that.
Let's be sure ignition switch is wired correctly now with the mechanic and the 5 blade new switch. THIS IS IMPORTANT.
1. Red wire should be plugged into ignition switch terminal blade marked BATT.
2. Pink into blade marked IGN
3. Resistance wire, which is either green or a funky looking wire, plugged into same circuit as pink.
4. Violet onto blade marked SOL
5. Brown into blade marked ACC
6. Red should show HOT all the time at switch.
7. Be sure you have the above wires to correct marked switch terminals no matter how you used the fifth terminal.
THE PINK WIRE IS GAS GAUGE HOT FEED. So mechanic was playing around near pink wire which, with a four terminal ignition switch, is ganged together with the resistance wire. WHICH IS THE ON/RUN WIRE!!!!!!! Which if not wired correctly, will not allow motor to run in "run" position. Then you changed ignition switch with a five blade. So this is a critical area involving exactly your problem.
After you have double checked above wiring, back to testing:
I will stay with this same simple test until it gives a definite result. Find your light tester. Clean the positive terminal on the coil. Inspect the two wires on the coil positive terminal for tight and check the crimped ring-tongue terminals to be sure they are not half broken off or wire come loose from crimp.
Then be sure battery is at 12.6 volts, or near, using your volt meter/multimeter. Then use the ice pick tester with black lead on the battery negative post and the red lead on the actual coil terminal stud and see if light lights up. There is no "bouncing around" with a 12V test light test. It either lights or It doesn't. Do this test with ignition switch turned to the "ON" position. Report.
Another test you can do is take off both wires at the coil positive and tape the ends. Install a test wire from the battery positive post to the coil positive terminal. Use a clip for easy removal. Start the motor using the key as normal. If it starts and runs, remove the test wire from battery to stop motor. Report.