Yep, that's a problem.
Can you tell if carbon is holding the intake valve open? You might be able to clean it out but from the looks of things, it may be time for a rebuild or a new engine. Or new mower. Looks like it burns a lot of oil.
That's a lot of carbon on the piston and valves. I would imagine the valve guide or rings (or both) are shot.
You're right in that you have nothing to lose by digging deeper at this point. Have you ever overhauled a small engine? You could use a few specialty tools, such as ring compressor and valve spring compressor but some people can get the job done without them.
As a plan 'B', you might check out the Briggs website and look on their 'repower' section to see if there is a bolt-up replacement for you engine.
Lots of small engine sellers online, too...Tulsa Engine Warehouse, Jack's Small Engine, Surplus Center are a few.
My Push Mower:
Yard Machine; 4HP 22"
Engine:
Briggs & Straton
Model: 10A902
Type: 2171B1
Code: 00032955
Air gap: .010
Plug gap: .030"
Last time I ran it - this year - it was hard to start, but seemed to run okay and finished lawn (didn't die, as I recall).
I removed air filter - for testing purposes.
I push primer bulb 3 times (gas squirts) - and engine only pops 3 or 4 times.
I spray starting fluid in carb and the engine only pops 3 or 4 times.
I remove plug and spray fluid directly into plug hole - same result
I have spark.
Dip Stick - shows oil full.
New spark plug.
New / rebuilt magneto
Cleaned fuel tank assembly.
Cleaned carburetor.
Replaced carburetor gasket and diaphragm.
So, I removed head - see photos:
View attachment 32365
It will be easier to see the difference if you open each image in it's own window and view side-by-side.
View attachment 32364
- only one of the valves opens and closes when I pull on the start cord. Is this my problem?
Once you get all that rusty crap off the crankshaft, the crankcase should slide off without too much trouble.
Keep in mind, other parts may come off with it, such as cam and oil pump. Take a peek inside now and see if it looks like other stuff is coming off with the crankcase.
edit: Don't use a puller on this unless you absolutely have to. You'll be risking damaging the crankshaft "bearing", which is nothing more than a bored hole in an aluminum housing, if you do.