Is cutting away the grass around the tree trunk a good idea?
It seems that most lawn maintenance companies use this method, and then cover the bare ground with mulch. Sometimes I think it's just easier for them. Then they have an easier time of cutting your lawn.
Our service on the other hand, seems to have cut away the grass, and NOT put down mulch, or any kind of ground cover. I'm in the Chicago area and we're going through a terrible drought right now.
In the future, would putting some kind of foliage such as Queen Ann's Lace, or something to hold in the moisture for the roots would be a better idea then leaving it completely bare soil.
I think mulch is a good idea. Your lawn care service probably charges extra to do it.
You can put down some newspapers on the grass and spray them with a water bottle to keep them weighted down. That should take care of the grass. Then put down some mulch on top. Not too much though and don't let it touch the trunk of the tree. You can sometimes get free mulch from your city/town. I've found pine needles to be the most effective at keeping weeds out but it has to be refreshed often. I used to pull over on the side of the road and scoop up pine needles with a rake.
Unlike alternative ground covers, mulch isn't going to need any water.... good news for the tree.
Part of the reason for mulching is to make it easier when mowing - to prevent the accidental chopping down of a small tree.
The other part of the reason also helps protect the roots by helping to retain moisture, as you mentioned.
I haven't heard any reason why plantings around a tree could be bad, except for the obvious competition for moisture and nutrients. A large tree could probably handle this, a small one could die, and a seedling would almost certainly lose to the faster growing, shorter lived plant.
Although I have wild blueberries growing up around every pre-existing oak out in the open lawn (and most of the others as well...), I clear a space around all the trees I plant and use a hardwood mulch to keep the competition out of the way. In about 50-100 years, I plan on letting the blueberries and other native vegetation grow around them as well...