Solution/lionel 1946 #671steam engine

I remember your post.  Did you replace the wiring?  I have a 671-RR, which is a 681 without magnetraction, so the guts are a little different.  I did a complete rewire of mine because the wire coating was starting to fray.  I also ended up replacing the e-unit.

Not sure which gear you're referring to, but by pressing it are you causing the brushes to make contact?  Could be too much play in the shaft and/or weak brush springs.

Have you used a continuity tester on all of the connections?  Maybe there's a solder joint that looks good but isn't, and it makes contact when you press. 

If I remember correctly there is a thrust bearing/washer at the motor shat where it engages the gear box. If that bearing/washers is incorrect it will lead to the issue you are describing. Compare what you have with the correct Greenberg parts diagram.

What RJT said.
If you Google search "lionel 671 exploded view", you will find the exploded view showing where the thrust washers go.  If memory serves, there are at least 2 and they could be different thicknesses.  You may have to reposition things more than once. 

If that doesn't work out, try taking a video of what you are seeing and post it here.  I too am confused about "I put my finger on the gear and push forward, the motor starts".  I am not sure where you are talking about.  These old machines are so simple, it's not likely this is unfixable; possible but not likely.

Tony V