I don't believe turning on the key will make a mechanical fuel pump make the fuel flow without the engine running.
This is why we continually beg people to supply the FULL MODEL numbers from mowers & engines because as I have never come across a FR 691 with a mechanical pump I would have given you the same answer and I have installed better than 50 of them as replacements because I used to get them cheap from RGS
So in my shop I would have first checked both the carb to manifold & the manifold to head bolts were tight
Then I would have gone to mow with a trigger pack of WD 40 or similar with me
When it started to play up I would have saturated the carb to manifold joint then the manifold to head joints , listening for a pronounced change in engine note
Next I would have plugged off the vent tube to the manifold , on some you need to remove the blower housing to do this , again a lean air leak s suspected
IF none of these made any difference then I put a 1/2 gallon of fuel is a substitute tank and hook it up to the carb high enough to gravity feed
If still no difference then I have some plugs that replace the fuel solenoid or an old one with the pin chopped off .
Still no joy then I put a pair of in line spark testers on the mower so I can visually see if there is a difference when the engine starts to run badly
Still no difference I shoot the mower .