Transforming pesky mulberries into friends

I have a number of mulberries that were never cultivated. They are tall, gangly, and in a stand of a bunch of weedy maples. I’ve tried the fruit and it’s fine, but there’s no way to collect it. It just falls on my kid’s toys and attracts flies.

I’d really like to turn these into helpful trees. One of my neighbors suggested cutting the trunk about halfway down and letting it grow new branches that are within reach. Even if this does work, I’m concerned that the tree will get shaded out by the weed maples. I’m no arborist. Thoughts on how to turn these pesky trees into friends?

If you have another area where you would like them to be, like not all over your communal area so falling on your kid's toys & attracting flies, and you are willing to wait a few years for new berries, mulberries (most & I'm not an arborist either) will grow readily from softwood and hardwood cuttings.

I had a box elder and a mulberry overpowering a pear tree , so I chopped both back to ~8 feet.
I maintain the mulberry by forcing the new branches downward, they "break" and then grow in the new position.
The box elder branches just break off.

I would fell the maples and pollard the mulberries to put all the branches within arms  reach.
This would probably pause the fruit production, but when it comes back it will be much easier to harvest.
The maples will probably come back from their stumps, but that can be dealt with pretty easily.