Need more info.
But I'll try to help anyways.
To be clear when the car doesn't start does the starter do anything?
If if cranks is it just really slow?
Does the engine turn over at all, can you hear the starter solenoid click at all?
Is the car a 5 speed or cvt? Does it use a regular key or push button start?
If 5spd, is a floormat jammed up behind the clutch pedal?
Iirc we had a few 2017's not start one winter, towed them to the dealer and they replaced a relay or two under the hood. Or so the dealer said.
I've replaced a few starters over the years, but our cars are started dozens of times a day so will/should wear faster than normal.
You could try tapping on the starter while someone tries to start it. Don't go to crazy and hit it hard enough to smash the case.
The starter is at the front of the engine at the bottom, behind a little metal curved skid plate looking thing. Need 12,10 mm and maybe a 14mm to remove. Unhook battery first unless you like amature welding or burning your car down. Ideally just remove the negative post. No need to pull the positive, plus negative cable is easier to remove.
For the shake at 60, the cheap and easy thing to try is rotate your tires front to back then see if shake goes away or moves to the back? If not your rotors are likely insanely warped, caused by either driver or sticking brakes.
Mirage videos:
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View my fuel log 2014 Mirage SE wussie cvt edition. 1.2 automatic: 37.7 mpg (US) ... 16.0 km/L ... 6.2 L/100 km ... 45.3 mpg (Imp)