Yes, the gears need to start and end at a certain position so that a tape-handling cycle can execute properly, then be in position to cycle again.
Since you've come so far and since you can do this close work, try flipping the worn gear upside-down. The reason is to engage teeth that are still in good condition.
Or though unlikely, you may exchange it with a similar (newer-looking) gear elsewhere in the machine.
Or, you might fabricate a replacement gear by making a mold of the original. Sculpt the mold so teeth regain their like-new full shape. Then cast a new gear made of epoxy, or plastic, or steel epoxy, JB-Weld, etc.
There are commercial kits used to duplicate a key. You get a vial, hardening gel, and metal epoxy. Immerse key in gel, wait for it to harden. Slice gel into two halves and remove key. (Sculpt the gel so teeth are shaped full and complete.) Pour metal epoxy into the empty space.
- - - Updated - - -
I try to playback on friends betacam BVW-35P and I got only audio with no video recorded on tape!?
Did you test that a different tape plays okay on his machine? Press the correct source button? The correct output button? Then immediately try your tape?
I don't know about camcorders but working with VCR's and tv's I find that everything has to be right. Buttons, settings, menu options, channels, etc.
And of course cables. One bad internal wire halts everything. It's an obvious thing to look for.
Video heads are tiny. There may be 2 or more on the drum. Examine with a magnifying glass.
You're probably aware that a slight buildup of dirt blocks the signal. Head-cleaning kits are available. The preferred tool is chamois. It does not shed foam particles or cotton fibers.