You have to observe the linkage in action with a helper pushing to see if that is the problem. There are all kinds of ways to "adjust". Sloppy/worn pivot points, worn levers or loose cotter pins would be what to look for. If you feel it is the switch or the button on the starter case. Both are easily available.
The question to ask yourself is what was the difference between when you "worked the linkage" under the hood and "pushed down on button on starter only". What was preventing the linkage from doing what your hand did? You need to test each of those methods you did several times to get an accurate result. To prove that one does not work and the other does, every time for 10 times. Then decide on a fix. So far, it sounds like the "stroke" on the foot mechanism is not enough anymore. BUT that one test you did is NOT enough to eliminate button or switch or battery or wiring.... yes batteries can show 12V static and still be bad.

I general for any problem:
What most advisors try to say as they guess what is wrong: Don't run around buying/rebuilding things until you know what is wrong by using simple troubleshooting techniques and observation. This is what most of the experienced members here do. Some just know what it is right off.
Sometimes it is easy and inexpensive to use the replacement method especially if a part is old. That's OK. It's just more rewarding to figure it out if you are not in a hurry or stuck. But that method can go on and on.


Right is right, even if everyone is against it, and wrong is wrong, even if everyone is for it. - William Penn