The recoil starter on my snowblower seems to work fine when I start it after pushing it out of the garage. But using the recoil starter after it's been outside blowing snow for 45 minutes does nothing. It just pulls out easily without engaging.
The recoil mechanism will be bolted to the shroud but you can probably figure it out without taking it further apart, as far as what little spring loaded thingy pops out to engage the flywheel.
On a snowblower, it's pretty important to keep the recoil well lubed. Not necessarily for the lubrication factor, rather the lube will repel water. When moisture gets in, it'll freeze and disable the recoil.
Well nothing is simple. Tried to take off the recoil and it seems to be held in by nuts threaded to a bolt with the head on the inside of the housing that covers the flywheel.
The nuts were tight and with some force just spin the bolt.
To get the housing off I need to remove some bolts that look rusty near the head.
On my McLane edger, if I want to get the recoil off, I gotta take the whole engine off because the frame member blocks the bottom of the recoil housing. Not hard to do though. All have to do is take the belt off, unhook the throttle cable from the carburetor, and then just remove the four engine mounting bolts.
It depends what style recoil you have. Briggs has used very different recoils over the years.
From what you are describing, it sounds like you may have the older style that has balls in it. Then the oil they put in for lubricant gets old, it thickens up like grease, especially in cold weather. The thick grease keeps the balls from settling in their grooves to the pawl can lock onto them, and the recoil will free wheel.
If you have this style, after you get the shroud/recoil off, the actual, ball ratchet mechanism will be threaded on the crank (you'll see a square drive)