You have to carefully dig them out with a sharpshooter.

Used a machete and got this bitty seedling out. Problem is: I've got about 1,000 of these. They're like dandelions in my yard.
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The reason they spread like that and grow directly next to existing trees as if they are trying to outcompete them is because the fruit/berries are very yummy to birds and the birds eat them them when they sit in your trees while they go to the bathroom, the berry remnants come out beneath the tree where it can germinate.

You've heard the term "fencerow tree"? Hackberry is a fencerow tree. Where birds perch to do their business, there will be hackberries a-growin.

You might try mulching heavier and using pre-emergent herbicide designed for garden beds. I would not waste time on or risk the damage with post emergent sprays like roundup.

They grow right up into the root flare. DH has told me for years that they would die if I kept cutting them back, but I don't think cutting them back works!

I can't mulch the root flares.

Growing into the root flare of Rose of Sharon
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Growing into the root flare of Texas Ash:
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I pulled a hackberry from this root flare. The hole:
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Don't even ask about my fencerow..LOL

Question:

Will pre-emergent herbicide prevent hackberry seeds from sprouting?

(Pre-emergents are promoted for herbacous / annual type weeds, hence I don't know if they would work)

I'll give it a try, but wonder if they would have any impact.

I don't know. That would be a good question for a landscaper or landscaper supply house. It's entirely likely that it would do nothing to help.