I know little about the inner workings of a transmission. I would think, however, that a broken shifter fork isn't going to work again after you shut the car off and yank and tug on the gear handle. I would suspect something wrong with the shifter itself.
Do you have a volt meter? Charge the battery out of the car, let it sit for 8 hours and read the voltage. Refer to the chart. If the state of charge is good you might want to try a load reading. Crank the car and have someone read the voltage across the terminals. If it's 9 volts or less the...
I prefer the left and, with one exception, the cars I've owned with tachs had it on the left.
Left:
1967 MGB
1985 Mustang GT
1987 Tempo
2008 Infiniti G35
2012 Nissan Frontier
2021 Mustang
Right:
1999 Saturn SW2