@canadachickens
Anyone ever tell you you're a genius
The mower is running as I type!

Finally!!!
I took your advice & put the jumper cables directly to the mower cables, fired up the truck, & hit the switch. The engine turned, but not at full speed, & didn't try to fire.
I pulled both plugs, burned off any oil (though I saw none), shot a little starter fluid into each plug hole, pulled the foam air cleaner (which was dirty) but didn't replace it, sprayed a little starter fluid into the carb port, & hit the switch again; fired right up.
I haven't a clue as to which action worked, but
something did.
Just for chits & giggles, I reconnected the battery; started right up! I guess the battery's good, after all.
The mower's still running; I want to burn up all the gas left in the tanks from last season (which I thought I'd done last Fall after I was done mowing) & start all over with fresh gas.
I washed out the foam air cleaner (gotta love Dawn detergent) & set it in the sun to dry. Once it's dry, I'l put it back in place & see if I can get the grass mowed.
I talked to a small engine mech yesterday & arranged for him to come out Saturday, @ $25/hr, in case I couldn't get it running...I'll contact him again if I have no further problems.
Many thanks, my friend; you saved me a lot of extra time & God knows how much money, & I'm more grateful than you can ever know.