From my experience and what I've seen, garlic cloves are often applied to soup in un-chopped glove.
Overtime, the boiling will release the garlic's sweet to the soup without melting itself into the liquid or dissolving into tidbits that attached to the ingredients.
I have also seen people pan fry the garlic gloves to golden brown before dumping them into the water. I believe the purpose is to harden the outside of the garlic while getting more exotic flavor out of it.
Otherwise, I would chop/dice/smash garlic for dips and frying.
I've never seen the kind of garlic bagels spiceyokooko is talking about, but I am familiar with garlic bagels of the variety you're referring to. I don't usually make garlic bagels at home because I'm happier with simpler ones, but I have made some ages ago.
I've usually found most things stick just fine to recently boiled bagels, but for fresh garlic, you may find it more reliable to brush an egg wash (roughly equal amounts beaten egg/water) on the top of the bagels before sprinkling on the chopped garlic. That'll work almost like cement to keep most of the garlic in place after baking.
There's some possibility that the level of moisture in fresh garlic may make it harder to just press the egg-washed bagels into a bowl of chopped garlic, as you would likely do with seeds.
Also, I wouldn't expect that all of the garlic will stay stuck. But with the egg wash, most of it will.
As for the texture of the garlic, there's a pretty wide range of common results at the variety of bagel shops I've tried in my life; some are golden brown and some are near burnt. That's a matter of cooking time, mostly, but also the starting moisture content of the garlic.
I would expect that most bagel shops use bulk, pre-minced garlic so that they can save on a little labor. This is typically a bit drier than if you fresh chopped the garlic at home. Consider buying the prefab stuff and see if that gets you closer to what you want. If it does, and you prefer to do the heavy lifting on your own, mince the garlic and let it sit on the counter to dry a bit.
Another strategy altogether is to mince garlic and mix it with some oil, and smear it atop the boiled bagels with your hands. I think this should stick reasonably well, and will affect the texture of your result.
Since I'm not 100% sure what your platonic ideal is for the garlic texture, I can't say whether the oil-mixture or egg wash route is going to be a better choice, or some hybrid, but those are some options to explore that I think will get you closer to your target.
Best Answer
In short, yes, but how much would depend on how long it is held and how it is handled. Simply the act of removing the protective skin opens the clove to air and deterioration, but not enough I would be concerned if the cloves are used in a reasonable period, the sooner the better.
Now, the shaking in a pot will tend to bruise the cloves. In doing so, you are rupturing cell membranes and increasing the deterioration speed. Slicing, crushing, mincing, etc., you have just multiplied that effect. The more you break down the cloves, the faster it is going to degrade on you.
Think in part this way, if you can smell garlic, or see the oils, you are losing some of its essence right there, and you are exposing it to air which is increasing any bacterial or chemical reactions. I would stop at the point of peeling and retain whole cloves, not process further than that until read to use myself.
Also note, some people, if they are going to use oil with the garlic anyway, will go ahead and mince, chop, or slice and drop into olive oil. Be cautious with this! It will trap you garlic goodness in the oil, but it will also create an air tight low acid environment which is a danger zone.