What turned the garlic purple

Since it was pickled, I believe it was a reaction with the vinegar causing it to turn green. Harold McGee has a good article on the very issue. He also goes into some detail in another artical here. I have heard various factors that can affect it, like the presence of dissolved metals, and the age of the garlic prior to being canned. However, I don't have any other info on those...

I do believe it's safe, assuming no other signs of spoilage are there.

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.