What part of the States are you in, Bsroses? Different regions have different weed populations. Also, have you checked with your local cooperative extension?
It may actually make little difference, though. The only help for most weed bulbs is to dig them out.
I am surprised that they have "choked out" the roses. Unless totally neglected, weeds usually only weaken roses, and make the re-bloom slower, rather than killing them entirely. You may have to blame one or more of your weed controls for that, especially if you weren't obsessively careful in applying them. Which ones did you use?

Could be wild garlic or three cornered leek... as Tugb. says it makes little difference you need to dig out all the bulbs and will probably have to for several years to come.

Agree with Moon Grower, wild garlic or three cornered leek, the latter sold as Allium triquetrum, the former Allium ursinum,both a real pain and take over anywhere they've been planted. I'm still trying to completely eradicate these from a garden I've been caring for for 20 years - after the initial careful digging up and disposing of every single bulb (in the hundreds) including the tiny ones, you invariably miss some and they pop up the following year, when they need digging out again.

This sounds like a situation that has gotten out of control and who's got time or patience to hunt around for hundreds of little garlic bulbs- not me! I can't stand the smell of garlic. I would rent some heavy duty motorized equipment - a roto tiller or even a small tractor like a cat. Ace Hardware, Home Depot, Lowes, Sears and Rent-a-centers. Just overturn the entire garden with a tractor, then sift out the bulbs with a rake. Plant a new lawn. You'll still probably miss a few, but you can pull them when they sprout.

Oooh,, no, I absolutely do not recommend you use machinery, although I know stating this openly may cause some annoyance to some people, but it must be said nonetheless. Even doing it by hand necessitates stopping after lifting every spadeful of soil out and sifting through to catch the tiny little bulblets that will be there after you've lifted out the main clump. Using machinery to 'overturn the whole garden' is the most efficient way to ensure complete distribution of the bulbs across the whole area, the same way it would if you were trying to get rid of bindweed roots. I'd even say don't drop any of the soil you've moved with your spade or fork onto another area, it's bound to contain teeny bulbets, whether you're aware of them or not.

I'll freely admit doing it properly and rigorously is incredibly tedious and time consuming, but if you do, you'll reduce the population of the bulbs down to .5% or less. I think another member on here had a bit of success using glyphosate - I seem to recall she crushed the foliage a bit, then applied glyphosate, and I think the member concerned was Seaburngirl, perhaps she will confirm or deny!

Totally agree with Bamboo on not using machinery... unfortunately it is a recipe for disaster.

I disagree with Bamboo and Moon growe. If you go back and read the post, the garden is ALREADY over run with garlic bulbs. That's what the post is all about. We are there already - even distribution over the garden due to years of neglect. If you hunt around digging up each bulb, you are essentially overturning the whole garden anyway - creating a bunch of puck holes.

Obviously Moon growe and Bamboo have never used a tractor but it's not difficult to learn. I have. A little training to properly operate is a valuable skill and will give you beautiful results quickly and efficiently. They are powerful yet can perform very delicate tasks.

They wont crush or break up the bulbs, but cause them to surface so you can see where they are. Then you can just pick up the bulbs using a rake or garden fork. This is why tractors were invented - they are quick, efficient - if you use them correctly. Just like anything else.

I just noticed the little addition in small print under your question Bsroses - this is a UK based site, and the majority of us do live in the UK, but we do have some American members, one of whom answered you initially, along with the member who's answered above me suggesting the use of machinery - he thinks you're in the States too. Your membership is displaying USA as your country of origin, hence the confusion in the answers, but you seem to be saying, in your addition, that you are UK based - it might be good to change your location under your profile to UK.

Dear Bsroses: I only focus on the essence of your question - getting rid of the garlic bulbs. No need to reveal your exact location if you do not wish to. It's not relevant to the question.

The questioner's location is already stated as Kent anyway,but it still would be best to change it in the profile so it doesn't show the USA flag.

Sorry, Bsroses, but I didn't notice the addendum with your location until now. As Bamboo and others have said, it would be helpful to edit your profile to reflect your actual location. If nothing else, it will help avoid confusion on future questions! :)
We don't have weedy wild onions in the desert, where I live (yet!) but it sounds horribly like a weed we do have called nutgrass. That is a corm forming sedge from India, and in my experience, rototilling the area just causes it to multiply, making a bad situation worse. When using glyphosphate on it, I mix it with a surfactant, to increase penetration in the leaves, and I spray just enough to wet the leaves, with minimal runoff to the soil, to avoid future problems with soil contamination.
Best of luck to you!

Tugbrethil Nutgrass is nothing like wild onions. It is a perennial grass which spreads via underground rhizomes. It's also very resistant to herbicides. Also, nobody has "weedy wild onions." and it doesn't sound like a weed.

The question is how to get rid of bulbs that have spread throughout the garden over several years. Tulips and Daffodils do the same thing. The bulbs have to be dug up. You can choose the method. Dumping chemical herbicides will preclude anything from growing there or reseeding the lawn.

Clearly one of our members has never seen, nor had to deal with Allium triquetum nor Allium ursinum, both one of the most invasive and pervasive plants you could ever wish to come across, and one of a few 'weed' plants that have comprised the bane of my life as a professional gardener over the years.

Nutgrass seems like a really good comparison, frankly.

However, I am fully aware that many people prefer to hang on to an opinion or idea despite any factual evidence to the contrary, and they are, of course, perfectly entitled to do that. In the end, Bsroses, the choice between conflicting advice is yours to make (though from the note added to your question, you've already decided to carry on digging out anyway).

Well the FACT is that Nutgrass, Allium Ursinum, Allium Triquetum and the bane of your life has nothing to do with the question put forth. You can simply tap "Home" key and read it for yourself. What more proof do you need. It's not difficult.

Most bulbs will naturally multiply and can take over a field if left unchecked. The identity is unknown. Nutgrass is resistant to glyphosphate, onions and most bulbs are not. You can't compare the two.