Starter is new and won't engage the flywheel.

Return starter bc the starter solenoid is not engaging bendix, which engages the starter teeth into the flywheel..

Check timing belt first , check starter also .

some starters require shimming for proper contact.. also check the flywheel teeth and see if they are ground off or broken(missing)

Doesn't matter. The forks inside the starter may be missing or broke off or bad spring in starter that pushes the gear out. Try new started. Starter may have to be shimmed to make contact with flywheel. Starter may be kicking out but not engaging flywheel.

check your flywheel & starter, one or both will be missing teeth, or the starter isn't "throwing out" yo engage flywheel

Fixed it get a flywheel shim .0040. For the 2.9.v6 amazing it works like a brand new engine. 5 starters 3 flywheels and all it needed was a shim that is made for that engine.

If the starter is spinning when you try to start but doesn't spin the engine over then you most likely have teeth missing off the flywheel ring gear. The facts that you say you have to push it and then it starts means the flywheel position changes and the starter has good teeth on the flywheel to engage. To confirm this next time the engine won't crank over remove the starter and look inside at the flywheel for missing flywheel teeth.

you have a faulty starter solenoid on the replacement starter or a broken flywheel teeth

If you turned the flywheel with a screwdriver and the bendix engaged the flywheel, then it sounds like it could be the flywheel. There may be "teeth" missing on the flywheel, causing the bendix not to grab onto the flywheel. Have the flywheel checked. If there are "teeth" missing, then replace the flywheel. I experienced the same problem with my Nissan.