If I was starting out, I would remove the plug(s) and see how easily the engine turns over. Smooth, effortless, turnover? Then it's not the engine.

The next thing I would do with the plugs out, is to apply 12V directly from the battery to the heavy terminal on the starter. If it cranks normally, then it's on to the solenoid. If the starter grinds, put the plugs back in with the wires OFF and try again.

Don't forget that the first thing you should do is to make sure the ground wire is secured to the new battery and the original ground.

Remove whatever seems appropriate to get to the starter. Maybe even the shroud, too. Just make notes as to what went where, especially, the screws/bolts.

Likely, it's something simple.

I think I would remove the spark plug, then try to turn the engine by hand in both directions to see if the engine will even turn.
Sometimes the starter drive (aka Bendix) will fail when the key is turned to start and get stuck in the flywheel ring gear.
If this happens, you may only hear the click of the solenoid because the starter may not be able to begin rotating in this condition.
Anyhow, after you get the flywheel cover off you should be able to see if the Bendix is jammed stuck into the flywheel.

Your engine translates to a -
Tecumseh TVXL220-157245C

http://toprake.com/index.php?module=documents&JAS_DocumentManager_op=downloadFile&JAS_File_id=2

Here is a link that might be useful: Manual?

Can turn engine by hand. Removed fuel tank, engine shroud, and spark plug wire. Removed the positive battery wire from the solenoid and touched it to the solenoid terminal that leads to the starter. The starter motor spun, but the Bendix stuck in the flywheel. Freed Bendix. Reconnected positive battery lead to solenoid. Starter cranked briefly with ignition switch, but then failed. Tried bypassing solenoid again, but get nothing now. Does this mean the starter motor is bad? I think I missed something above about grounding. New solenoid is $15, if available. Starter motor $200, if available. Beats buying a new mower or paying someone else to fix this one.

Sounds like I'd check to see if the battery is fully charged.

ditto baymee's remarks. take battery to qualified shop that can load test it.

The battery is brand new. Was the first thing I replaced. However, I suppose that does not necessarily exclude it as the problem. Thanks everyone for your suggestions.

test it. "new" batteries of that type don't have a great record of being good new.

Sounds to me like the plastic starter gear on the shaft has failed, and is getting stuck in the flywheel teeth. note that the plastic gear is not to be in the flywheel teeth all of the time! It spins up to connect to the flywheel gear teeth, to turn the engine crank-shaft, and when the engine starts, as you allow the ignition key to flip back to the run position, the gear falls down away from the flywheel gears, and the engine is running.
I wonder how much money has been spent by the uninitiated buying new starters, when a starter shaft gear costs only a few bucks.
And, i had one man ask me to look at his lawn tractor, an older one with a B&S engine. Said he bought a new battery, and a starter (used one), and that would not be the problem he was having. I saw he had the battery connected the wrong way, so i switched the connections at the battery and it started right up! Somebody should print up a "Trouble-shooters guide" for those who can't afford to have a shop look at it! But only sell it to those who know what "Lefty-loosey/ Righty-tighty" means in mechanic talk! And, how ya apply it when laying on yer back and looking up at the part!

I might go ahead and remove the starter, doing a few dry runs direct with the jumper cables. (touch the ground to the starter housing last as it will spark- batteries and sparks don't get along.) The starter could be dry or the magnets could also be letting go inside. These starters weren't known to last 4 ever. Hopefully you've covered your bases and cleaned the terminals in the starting circuit well.

One other thought -- inspect the "threads" on the bendix. You may find on close inspection that they're damaged. Look for fresh plastic surfaces.

I went ahead and replaced the starter motor and she fired right up on the first crank. I appreciate all the advice and tips. I probably could have dug into the starter and saved some money by repairing it, but I ran out of time and patience. Again, much thanks and I give this forum high marks for helping me solve this problem!