SDS drills wobble a bit due to how they are constructed.
The drill is actually secured by two ball bearings and can freely move in and out. The whole purpose of the SDS connection is to keep the bit from falling out and mostly centered, while the actual "hammer" part rams it into concrete (pneumatic or oil operated).
The drill bit hammers into a wall like a chisel.
It doesn't have to be sharp, just sturdy enough not to bend. The only reason it has grooves and rotates at all, is to efficiently remove the powdered debris from the hole. Precision is not a big deal here - the concrete does not chip accurately, and the smallest sds drill I saw was 5mm thick, so being off by a millimeter or two does not matter.
Now compare it to regular drill, where the bit is rigidly screwed into a chuck. It cannot move to the sides and should never move to the sides (if it does, replace the drill).
The drill does not "hammer", it shaves.
HSS drill bits cut the surface. This tool is designed for precision drilling into metal, wood etc. You can get drills starting at 1mm thick. The 'regular' drills often have the 'impact' function, because manufacturers want to brag "it can drill in concrete too", but they always fail to drill a hole in any kind of concrete. The "impact" functionality is just a gear VS the pneumatic operation of actual impact rotary hammer. And you don't want to accidentally impact-drill with metal or wood.
From the points above - when you drill into metal, wood, plastic, etc. use a plain drill with a chuck, preferably get one without the "impact" function. It doesn't have to be powerful - 200-300W is absolutely fine. Even a battery operated will do. I prefer cheap light drill for working with wood, so I can easily hold it with one hand.
Forget about the "impact" mode and conrete - this is not the tool for the job.
Want to drill into concrete? Break out the big boy. A proper sds rotary hammer with pneumatic or oil impact mechanism, 3J or more impact power, and you can get through everything used in domestic building. 500W minimum, 700W recommended. Just make sure yours has a clutch, or it can break your arms when it gets stuck in rebar. Battery operated? Please excuse me for a minute while I laugh my butt off.
(Tangent advice: Hold it with right hand on forward grip and left hand on the trigger. This way, when it gets stuck, it will jump out from your left hand and you let the trigger go. If you use right hand on trigger, it will ram into your hand when stuck, and your finger may keep pressing the trigger which leads it to keep spinning, potentially breaking your hand. Trust me, it hurts when it jams)
You may notice I did not mention the chuck adapter above, because that idea is an abomination invented for people who think they can do all the job with one tool. The adapters exist, but they will be wobbly by nature. Just use the proper tool.