No useful purpose other than sling attachment. It seems that it would be detrimental to accuracy.

Leaning towards leaving it off of a fresh green mountain barrel install.

Interesting question.  Why don't you shoot for accuracy with and without the barrel band and report the results here.  I, for one, would be interested in the results.

The one I had bought in 1969 with the barrel band was far more accurate than any I have shot since.  While free floating a barrel can be a great idea on centerfire rifles the distances one uses a 22 at I don't think it makes a lot of difference. I have watched some youtube videos on 22 and 22 mags in the precision rifle build from Ruger and my conclusion is they lack significant precision, or the shooters totally sucked.  The old saying was "Ruger who needs accuracy" but just like Remington's they can be made to shoot if you throw enough money at them. They are however reliable, every Ruger I have  encountered would go bang reliably. Can't say the same for Bergara's version of the 10-22 with 3 fail to fire in the first 30 rounds and a trip back to the factory.   Your new barrel is more likely to improve accuracy than anything else. Often just having the crown on a barrel re-cut by a competent gunsmith makes a very noticeable difference in accuracy with any firearm that shoots a loose group.

The barrel band is often discussed. My experience with them is they don’t matter. Even if you use one as your sling attachment point and put pressure on it using the sling as a shooting aid, it doesn’t matter. The thing that seems to make the most difference as far as accuracy is making sure the screws that hold the barrel block on are tight and making sure the action screw is tight. After that it’s using ammo the gun likes and the fundamentals of marksmanship.

YMMV so it might be worth trying the rifle with and without the band, but usually it won’t effect accuracy.

You have to shoot it with and without the barrel band to see what's best. I took it a step further by doing that plus the free float option. In order to do a quick free float I made a spacer out of a piece of .025 brass stock to drop in the stock where the stock screw is and shot some groups. My gun got about a 20% reduction in group size with the free floated barrel without the band. I like the looks of the gun with the band plus I wanted to keep it as a sling attachment point so I put the small drum sander bit in the dremel and removed material from the inside of the part of the band that goes over the barrel until there was daylight between it and the barrel when everything was put back together. You can have it both ways if you need it but you have to test all the options to see what works for your gun.
One of these days I'm going to send the barrel to Randy at CPC & have it rechambered. He removes some metal from the breach end and rechambers it, recuts the extractor notch and
Shims the barrel v block dovetail the necessary amount. Everyone that's had it done says it dramatically shrinks groups.

I was thinking about the barrel mounting and I can see where the band transfers stress to the stock instead of the barrel locking clamp. bump the barrel down and it hits stock, bump the barrel upward and the band transfers the stress back to the stock again... So I guess it would be better to keep it. Taking a few thousandths off  the inside of the band to give it a slight free float might be the best of both worlds as mentioned above.