You don't need to have a bare perimeter around it at all. There are lots of fast-growing, shallow-rooted plants that could work. The question is, do you want something that covers it year round, or do you want something that looks really great for the height of the growing season?

No matter what plants you choose, the most aesthetically appealing solution to deal with year-round ugly problems would be to make a little wooden trellis "cage" for it. You can just get a couple of 1x2s, nail them together like a box frame, and just screw on the trellis pieces, leaving it open in front for access to the door. (That you can cover with some vines trellised in a container - if you use the same stuff, it'll blend seamlessly.) That way, no matter what the season or what you decide to plant, you'll be dealing with a charming wooden lattice structure instead of an ugly metal thing - much easier to incorporate into the landscape. Making one would require zero carpentry skills, about two hours, and little more than a saw and a drill with a screwdriver bit. If it's not perfect, who cares? Bob Vila isn't coming by to inspect your utility boxes.

If that's too much effort, you could strategically place large pots with short trellises around the whole box and/or on top of it, and plant them with fast-growing annual vines like morning glories or moonflowers (and maybe mix in sweet peas for earlier bloom.) You could also cover the box in pea netting or chicken wire around everything but the door, so you the vines would grow over the whole thing.

If you don't like vines, I'd think about would be flanking it with tall, well foliated plants so it's masked altogether. It sounds like you have that going on a bit with the grasses already. ...Not for nothing are hollyhocks also called "Outhouse Flowers." Gaura is great for that too; you could try something like nicotiana or hesperis (I'm not sure how deep the roots are on those though.) I'd plant a variety of things so you had season-round coverage. This option (unless you do the trellis box) is still going to leave you with something ugly in winter.

If you really wanted to do shrubs to cover the boxes, I'd recommend flanking the box with arching antique roses that'll all but cover it up, then pruning them to leave access to the door and erecting your trellised container vines just in front of the access door. There are a lot of very lush, well-foliated antique hedge roses (like Albas, Rugosas, and maybe Portlands) that bloom a ton, will cover up what you need to cover, and then offer beautiful hips all winter. This would look especially nice with the trellis-box to hide the ugly metal, but it'd work without it too.

...Roses excepted, though, I'd personally skip most shrubs. They don't provide cover as well as you'd hope, they take up a ton of room, and their root requirements are going to go much deeper than a most annuals and a lot of perennials. (But I'm very protective of the space in my tiny yard that might be taken up by relatively short-blooming shrubs - so take my opinion with a grain of salt for sure.)

Let us know what you decide!