Appears you have at least two problems with that rig. (hold off for now on getting the goat as a mower)
First problem is the throttle cable linkage is not getting the full range of motion to the carb.
Whirly has seen this also. Appears somone has dicked with the throttle assembly in the past maybe that you are not aware of and the engine has been just getting by ok while you had it.????
Next once the engine starts it's starting to stutter and finally die. (first in your video the engine is being cold cranked and started with no choke applied and most all Briggs engines of your type require a choke to start easily and readily,
especially when cold and some even when hot re-start.
In the first video it's apparent that the engine is being started with no choke applied.
If you are moving the throttle to choke the metal lever IS NOT PUSHING AGAINST THE choke linkage wire. The choke linkage wire should be pushed forward by that lever by about 3/8 to 1/2 inch. This is normally adjusted by loosening the throttle cable clamp by the white plastic piece and while the throttle lever is fully forward on the dash slide the throttle cable to the rear until you get some choke applied
then re-snug the clamp. If you completely remove the throttle cable wire from the plastic piece
and move the plastic piece to the rear and you don't get any choke something is wrong with the linkages at the plastic piece assembly. If you remove the throttle wire from the plastic piece and slide it back and forward with your fingers and see full range of motion in the carb throttle linkages but you cannot get full linkage range adjustment when you re-connect
your throttle cable you will most likely need a new throttle cable. (a possible cause of this is because someone has re-bent the wire and it's too short.
In your second video the engine sounds like it's beginning to stutter.
I'm asking you these questions so as to try to determine i
f the engine is getting to much gas (flooding) or a weak ignition.
Do you see any soot or smoke from the muffler that would indicate the engine is flooding when it starts to stutter?
Is the throttle lever still at the fast starting position when it start to stutter?
do you see any smoke from the muffler after it dies?
Do you see any sparks from the muffler while it's running?
Remove the spark plug and Post a picture of your spark plug tip?
AND just because a carb looks really clean (as you say) is not a indicator that it is good. I have some carbs that I use a gravy or soup bowl and they are bad.
You need to charge your battery before you ruin your starter.
Answer the above questions and give us a report back? If the mufflers exhaust is sooty do a video of it.