[kaptain]

First of all.

Contort yourself so that you can see the backside of this valve. It is possible that the nut which secures the assembly to the console is loose. So when you try to push it in, the slack in the mount allows the knob to hit the console instead of pushing in far enough to engage. When you pull it out, this is not a problem.

Easy diagnosis would be to put one hand on the back of the switch assembly and push the knob in with the other hand. You shouldn’t detect any back and forth movement.

The loose nut issue HAS been an issue on other Newells of this era.

If you need a new switch. https://www.napaonline.com/en/p/MBIKN200...lsrc=aw.ds&

Good point. Thank you very much, Richard.
It does look like the push-pull valve's metal mount lacks solid support (see photo). 
Once pushed hard to release, the yellow knob (almost? or already?) reaches the leather panel. 
As for the replacement part for this valve, the one shown in Richard's link appears made of plastic, but the existing one in my #701 is made of metal.

Because the yellow knob says "Bendix", I am trying to figure out which Bendix valve it is