I do not agree with the suggestion of making Mage Hand concentration. There are already a lot of other spells which require concentration. Even with the so-called combat applications which it really should not have in the first place, it's not powerful enough to deserve concentration. I would sooner get rid of all the combat applications to it before making it concentration, and I think that removing the combat stuff should probably be done anyway.
I do agree with the maximum range being from the caster, not as far as it can fly in one minute. It's not a scout.
In combat, the Mage Hand should have the same initiative as the caster. It should only be usable if the caster has a bonus action, and using the bonus action consumes the mage's bonus action. Obviously the programming side of that would be tricky to implement.
I think Mage Hand should have an Inventory which can hold exactly one item and that can weigh no more than what, 5-10 pounds at most. This lets you pick up a thing and move it. If the hand ends, the item drops on the nearest valid solid surface to the hand's last position (so you can't lose quest items into bottomless pits.)