***" Should I take the top cover off and attempt to turn the engine by hand?"***
No, not yet. First, take out the spark plugs and try cranking the engine with the plugs out.
If the engine will spin when the plugs are out, the main problem might be the need for a valve adjustment so the compression release feature can function properly.
But that only covers one problem area.
How come you just keep on relying on a boost to get the engine started?
Don't you think it's about time to fix that too?
How old is that battery? Couple of years or more old battery is suspect anyhow and with your admission that is has required a boost each time you want to start the engine makes it even more suspect.
And, the fact that you say it runs for a shorter and shorter period as time goes by tells me that you likely have a faulty charging system.
You can enhance your forum consultation by posting the Briggs Model Number and Type number from the engine, and the full technical model number (and brand). Technical model numbers are usually under the seat on a frame decal.