While I'm stuck in this hot, stuffy room, I had to figure I might as well try and work on some of the Unfinished Objects in here.
It's too late to put the catbed quilt to its intended use, but on the other hand the larger intent has always been to use it to test things out for the grandma quilt, so, onward.
I'd ordered a free-motion quilting foot set and a super slider some time ago, and finally got to work setting them up yesterday. I think my overall conclusion is going to be that I will gladly pay someone else money to quilt the grandma quilt. Here's basically what I got done before I ran out of thread:

I still haven't gotten the correct tension for quilting completely dialed in on my machine. Evidence:

This is "eyelashing," where the bobbin tension is so much higher than the top thread tension that it pulls extra top thread all the way through.
By the end, I'd managed a couple more adjustments that improved things further. Not perfect, but at least acceptable. This is the backside of the green fabric in the first image:

...but as mentioned, I've run out of thread for now, so this project is on pause again.
Frustratingly, although I attempted to order a free-motion quilting foot set from Janome specifically to ensure that it would be compatible with this machine (Janome HD-1000), a couple of important parameters aren't right. For one thing, the needle wasn't centered in the foot, which seems like a really basic thing to get right. I had to make a small cardboard shim to get it correctly centered, after having broken 2 needles. The shim is barely visible in this photo, as a tiny bit of blue-gray behind the needle, but seems to be working well now. Millimeters matter for this kind of thing.

The set also came with a clear plastic cap to cover up the feed dogs, which is what's visible through the opening in the white super slider. The cap snapped into place just fine on the machine, but if you look closely, you might notice that the needle is passing through the rightmost portion of the slit in the cap. It seems to me that perhaps the needle should be passing through the
center of the slit in the cap, but what do I know? So far, the feed dog cap doesn't seem to be causing too many other issues, so I haven't attempted to modify it yet.
All said and done, I can at least declare two forms of forward progress on this project. First, once I get more thread and sewing machine needles, I should be able to finish up this mini-quilt in a reasonable amount of time*, and perhaps someday soon there will be another cat or two who might use it (the way these things go, with cats). Second, this has furthered my resolve to find and pay someone else to machine quilt the grandma quilt for me. There just isn't much point in my learning how to free motion quilt just to finish the grandma quilt. I'm just not very well set up to do it, and the full setup costs and work are more than I'm inclined to deal with. This is a common practice among quilters, for good reason, and I'm really not looking to take up quilting as yet another hobby. I just want to finish the grandma quilt so it can have a life as a useful and at least somewhat aesthetically pleasing object.
*This is after concluding that hand-quilting it would require an UNreasonable amount of time, although the hand-quilting is lovely and pleasing in its way.