I used my mother’s old knee highs to support the melons.

Trellis was concrete reinforcing fencing.

Exactly - old tights and a lot of tying in. In the past, I have grown a lot of squash up one side and over an arch or in one case, an old bedframe (with lettuces underneath). The fruits mostly form when the plant is tending to the horizontal so if you can utilise some structure which has a horizontal element, it is much easier.

"The fruits mostly form when the plant is tending to the horizontal so if you can utilise some structure which has a horizontal element, it is much easier."

That is interesting - I wonder why? That may have been the problem, the cattle panel was straight up. Cukes set fruit like crazy on a vertical trellis, and they're in the same family.

Do I have to support the fruits with netting/hosiery? Or is that just an insurance policy in case of high winds or something?

That really makes no sense. Fruit doesn't care what direction it's growing in. Reference? I could imagine that a plant gets more sunlight when spread horizontally, though. But the parts of the plant down below get less.

I've found sometimes squash needs a little help climbing. Personal melons have too. Or at least by comparison to how easy cukes climb on their own. I support or not a bit depending on size and weight and how long it stays on the vine. Like a personal melon under a pound might not need support. One year I grew a summer squash vine and it didn't get supports. If it's something that gets heavier than that, I tend to support. Some vines are darn strong and the fruits didn't need it. Mini-butternuts got over a pound but didn't get support, though they probably should have. The luffa year those things were huge and the plant needed support, not the luffas. Tiny gourds didn't need supports. Some other winter squashes the weight of the squash would start pulling down the plant a bit, I put in a support. Just like I add in a support or tie up for a weighty tomato cluster as its ripening if it's pulling the plant.

What I mean is that the vine has usually put on some length before there is viable fruit, (because the first few are always a bit iffy) so by the time I am looking at squash ready to tie up and support, they are already getting towards the top of the vertical supports and heading over the top...where it is easier to offer extra support as the plants grow on top of the structures. I don't think it works like training roses or other woody plants, where the dominant vertical shoots are interrupted (by bending, pruning, pegging out etc.) where emphasis is on keeping laterals growing in a horizontal direction to slow down the sap and encourage more abundant (but smaller blooms, rather than one huge flower. Then again, maybe it might??

Maybe the bees/pollinators look down not straight ahead.

OK, so you're saying that if your trellis is too small, fruit will start to form once the vine has turned over at the top. That's a little different. I believe that it is true that sap does indeed flow more slowly when the vine is horizontal (that's how capillary action works), but it isn't clear to me how slower flow makes more abundant blooms. I believe it is the case that climbing roses (but not regular roses) want to bloom at the top of the stem, and if the stem is horizontal, the whole stem is the top. But my squash and melon blossoms really seem to be distributed pretty uniformly, so I don't think that rule applies to them,

Every squash I grow send sout roots into the ground at each leaf node. Sometimes I let it grow over concrete, gravel and bricks in my garden walk. Those vines never do as good. One year I have a massive zuccinni vine 40 feet long. I noticed the base where it originally came out of the ground looked rough. On closer investigation it was completely gone. Huge vine many huge zuc's and zero nutrients from original stem/roots! That was eye opening.

I suspect the missing extra roots are why.

I grow over a long run of landscape fabric, so no rooting along stem nodes in my case