Can't find sweet spot on the gear handle to engage starter?

It acts like you have burnt spots on the starter shaft that touches the brushes ( voltage) to start. When you rock the vehicle in gear the starter shaft moves and it is placed in a position where the brushes can make a clean contact. Then when you turn the key, the starter will engage and the vehicle will start. My recommendation would be to change out the starter.

When you replaced starter did you notice the condition of the
flywheel ring gear? My guess is the ring gear is very badly worn or somehow the positioning of the starter is such that it is off enough
that it runs into the ring inside the gear. Did you hit something in the road before this symptom started? Any chance someone tried to jack up the car and used the starter as a lifting point?
Accident? Are you sure it's your original engine? Ie they used to be very easy to change. Any other thing that would bend the mounting point on the engine? If manual transmission, try to move the engine a little by moving the car with clutch engaged.
If Automatic try to move ring gear with a tool if you can get to it.
If this action is sufficient to move the ring gear into a "good" spot
and the Starter will engage, you now know you have a few bad spots.

does it fail to start or crank, JUMP started?
does the rested battery read 12.6v exactly or more (warm day)?

battery dead? charge it.
battery cables terms filthy. (corroded)
Not in park? got A/T or M/T? can help not knowing tranny.
not holding brake pedal down (USA? where are you, 1000s of posts and not one post says, hey im in Canada or?
try wiggling the gear shifter.
try wiggling the brake pedal (harder)
try hitting the starter, yes, this can wake it up.
the starter is bad, you can hot wire any starter to test in on car, ask.
the key line that actives the starter is dead, ask how to test that.

I had a similar problem with my 06. It was solved without replacing anything. I depressed the starter button, THEN slowly applied the clutch lever , which allowed the starter to engage at its "sweet spot". After a few times of this process being effective, the "sweet spot" was re-established, and problem was resolved. Hope this helps.

You could have a bad spot on the flywheel. The flywheel is a plate with a ring of teeth the gear on the starter engages. If you turn the crankshaft a few inches and the starter works, I would check the flywheel.
Otherwise, it could be a mounting problem with the starter, or something keeping the bendix gear from engaging.

If you mean the starter grinds from time to time and won't crank the engine when it happens, the problem could be a bad spot on the flywheel and/or a faulty starter. The starter uses a magnet and bendix gear to push the starter gear into the flywheel. If the magnet or bendix is weak, the gear will not engage and can damage the teeth on the flywheel.
You need to have the starter removed and check the flywheel teeth for damage.

Becausethe starter drive gear disengages after each starting attempt, the engine ismuch easier to hand prop should the battery be too low to provide normalstarting voltage. Though we do not recommend had-propping an engine, it doesoccasionally still happen in the field.

If I understand correctly the starter motor turns but does not cause the engine to turn over. If this is the case then the ring gear on the flywheel or flexplate is worn out, or the teeth on the starter motor are worn out. The starter gear does not engage the flywheel. With the starter removed the teeth of the flywheel can be viewed. turn the motor over by hand and look for worn spots; the gear teeth will be noticably shorter. If you find worn spots then the transmission will have to be removed to replace the ring gear. Another possibility is the starter is not pulling in correctly, thus the gears would not engage.

Hold key in the start position and move or wiggle the transmission lever a little; you'll hit the sweet spot and it will fire!

The starter is actually an assembly with a motor and a solenoid. The solenoid does two things: it closes a contact to send electricity to the motor, but it also moves a gear on the motor shaft upward to engage a gear on the engine flywheel. Most likely, you will have to replace the starter assembly to fix this.
Just in case, you should also check the gear on the engine flywheel. It's where the starter pinion engages. If it's missing several teeth, you could just be in that 'dead spot'. That can temporarily be solved by turning the motor a little bit to get to a good spot - but remove the sparkplugs first! And eventually you'll end up in the dead spot again. It is a more expensive repair.
Good luck! Please rate if you find this helpful, thanks!