Before you start ripping the engine apart there are some simple things to check.
The first & most iimportant is to post your engine number and your mower number.
Then those who are in the trade & have all the information at their fingertips can give you the correct information for your machine.
Untill that what you will get will be just general guesswork which may or may not be what you need to know.
Next try some simple things, follow the throttle control cable down to the engine, check that the cable clamp at the end of the control cable is done up tight and the cable has not moved, preventing the throttle from fully engaging the choke.
The choke is as the air filter end of the carb & the throttle is nearest the engine.
When you see where the choke control rod is, move the throttle lever fully to the choke position and check that the choke control rod is fully on ( can't go any further).
If not loosen the cable clamp & move the throttle control wire till it dose, thighten the cable clamp then check the choke is fully off with the throttle in the run position.
And yes, unless this is a very new machine with auto choke, you do have one it is intergrated into the throttle control.
If the engine turns over very lumpy, then by all means check the valve lash however check around the rocker box for silicon sealer.
If you see some there then B &S did not fit a rocker box gasket so you will need to either buy one or get some gasket maker.
Ensure you remove all of the old silicon if it is there.
The external tap is a good idea, particularly if you fitted it between the tank & the fuel filter ( a contensious issue on this list ) and it will serve you well to turn the fuel off when you have finished mowing and let the engine starve but don't walk away & forget to turn the ignition off.