Toro's even the homeowner model are not inexpensive. Are you handy with tools? All things considered, if the deck, tranny and all the other non-engine items are in good shape and have a lot of life left in them, I would check out the cost of a short block. A short block is a engine, minus the carb, air filter, flywheel, etc. All those come from your engine and installed on the short block to make a complete engine?

It's not rocket science. You can do it in about 4 hours with a basic set of tools, a bit of patience and a few cold ones.

You're sure the starter pawls underneath the starter aren't broke and missing? A starter without a pawl will do the same thing. Pawls just run a couple of bucks.

When I have blown an engine there is no way I have
seen the motor turn over freely.
Usually a rod is going to jam up.
I would look at the top of the motor screen and see if
it turns when you attempt to engage the starter.
If it turns, next I suspect a sheered flywheel key.
Next remove plug and crank to see if piston moving.
You can even press thumb over hole and feel compression.
KEEP AWAY FROM PLUG WIRE and MAKE SURE THERE ARE
NO LOOSE PARTS IN THE CYLINDER.

Quick check - does the blade turn when you pull the starter? If so, the connecting rod probably broke off the crankshaft and the engine is toast. If the blade doesn't move, maybe the flywheel key is sheared as above, but the crankshaft could also be broken (very rare).
Then there's the issue of stalling during use; the engine may have been overheating due to chaff build-up in the cooling fins under the steel air shroud. Forcing it to keep running may have forced a blow-up. Of course, it may not be 'blown up' at all and the stalling may be fuel or ignition-related.
Let us know what you find.

The engine is definitely turning over when the starter cord is pulled. I can hear the usual wump-wump-wump of the piston cycling. But it turns over very easily and continues turning over for much longer than it normally would with the same force of pull on the cord (wump-wump-wump-wump-wump-wump-wump-wump-wump-wump-wump-wump). In fact, it feels exactly like turning over the engine with the spark plug removed ( I checked and it is not loose).

I talked to a relative who knows a bit about small engines. He thought the likely problem was a stuck valve.

I'll do some more diagnostics as suggested above and report what I find.

Very possibly a stuck or broken exhaust valve, or it would be blowing back through the carb. It could also be the cam or most anything else in the valve train. A stuck valve would be good luck indeed. Usually some Seafoam in the cylinder and several cycles of soak/wait/pull will work it loose. Consider adding Seafoam or Marvel Mystery Oil to your fuel from now on.

What did you pay for the mower??If it turns out not to be a simple fix then I'd seriously look at buying a new John Deere JS series mower or a Husky 7021RB.

skane1, Seriously? That is your "helpful" advice? Do you just copy and paste that into every thread? Come on.

Could be a stuck valve or it could be the camshaft is out of time with the crank. When you pull it through is it pushing air out the carb?

Seized piston and busted crankshaft per the Toro dealer. Not cost effective to fix at their shop.

Time for a new mower. Is the old engine a Tecumseh? There are usually plenty of used mowers/engines around. You could swap in a used engine and have the old Recycler as a spare/backup mower. I cannot imagine what happened to the engine - Tecumseh engines are very tough.

If your going to trash it - I would be interested in the side discharge chute.

Did you determine what caused the piston to freeze and break the crank? Was it low on oil?

'The engine is definitely turning over when the starter cord is pulled. I can hear the usual wump-wump-wump of the piston cycling.'

This does not jibe with the stated reasons for the breakdown.