Plastic anchors come in different quality flavors and having the collar or not doesn't correlate with that.
I use a locally manufactured (I'm outside of US) series of anchors without a collar - their diameter is slightly smaller than that of the hole and they have small relief dots extending from the surface so that the outer diameter is slightly bigger that that of the hole. They require some force to be driven into the hole - either pushing them with a hard tool or gently hammering with a hammer. My experience with these anchors is very good - they almost never slip given the material is hard and doesn't crumble while drilling.
At the same time I saw a lot of anchors both with or without collar that slip in the hole. So I'm pretty sure it's not the collar itself that matters.
That said another very important factor is how precise the hole is drilled. It has to be of exactly the required diameter. Drilling must be performed in such manner that the drill bit doesn't exert side load onto the hole walls - otherwise the hole is milled and the anchor won't hold there.
Also usual plastic anchors can only be used in strong solid materials that don't crumble while drilling. There're special "hollows anchors" (not sure of the exact term) that can be used for slightly bigger holes and for hollow materials - they tight into a knot when inside a hollow.
Finally the anchor diameter must be properly selected depending on the screw diameter. Otherwise you're screwed.
I actually really like those linked triple grip wall anchors (linked by OP). The package often comes with its own drywall drill bit as well. The trick to using them is to pinch the "shoulders" in until they are flush with the sides of the anchor body before pushing them into the hole. Drywall is fairly chalky and brittle, so it will give a little bit where needed to let the shoulders through. Once in, these are nice, strong, low-profile anchors. The auger ones are a lot more destructive and tear up a lot of drywall if they fail. But I've used those successfully in the past as well.
Best Answer
Yes and no. It can be removed but not without doing some damage to your drywall.
The trick is to get the nut at the rear to go back again. To do that you need to get under the front flange and be able to hold it and the screw-head together while you screw it back.
Once it's free enough to get some sort of substantial plate behind the flange you can remove the screw almost as far as it comes and hammer it back in. It should them pull out with a little bit of persuasion.
However, since you will ultimately end up having to repair the wall anyway, it's simpler just to push it through. Use your screwdriver as a centre-punch and hammer the screwdriver till the anchor falls inside the wall. If it is really stubborn, do not be tempted to just hit it harder. Cut or drill some small holes around it.