And the first sign of excessive valve lash is an engine that will not start cold in the morning right out of the shed but once hot and the gap closes a little will start & run all day.
However the cranking will be a little lumpy.
If the mower was on the workshop floor & it could be seen, heard & smelled they it could be diagnosed in 2 minutes flat.
However all we have to go on is words on a screen and as you should know, the first step is to rule out what it is not.
So very low compression from a sticking valve or worn rings will allow the engine to crank too fast but be difficult to start
And of course too much valve lash will make it crank lumpy as will a faulty decompressor or worn cam.
A leaking head gasket will also make the mower difficult to start but allow it to run fine once started.
In each & every one of these cases a shot of starting fluid will allow the engine to start, as will a choke that is not fully closing as you have already suggested which could very well be the actual problem.
When you go through the list of things that Deckeda has done in his original post the only thing missing was checking the valve lash.
So assuming that everything he replaced was actually good that does not leave much.
And as a "cute' example I had a Murray with a 17Hp intek.
The customer would ring me early in the morning and tell me "it wont start, I ,have cranked it till the battery has gone flat "
I would turn up several hours latter, hook up the jumper pack & it would fire right up.
So after cold one for payment ( he & his very large family are all very good customers ) I would be off to the next house call.
Next morning, same thing.
Now when I say early, I mean crack of dawn type early & the mower lives in a steel shed out back.
After a 1/2 dozen of these , I took had a loaner come available , so swapped the mowers & took it back for a service.
The valve lash was massive, the cam was worn almost round and the exhaust valve was just about burned from the edge to the stem.
When I asked him about the lumpy cranking in the morning he said he thought it was "normal" as the mower had always done it from new.