I find myself having to use checklists of really simple stuff (like going over all captures, checks, threats, tactical themes, changes after a move) quickly in order to avoid very basic oversights from happening. Having to resort to this is of course quite discouraging, and I'm asking myself if the necessity to do so disappears as one grows stronger. So, do strong players (let's say rated 2200 and above) use such checklists?
Disclaimer: Every person has their own way of approaching these situations, but I think my experience is similar to the experience of most other (stronger) players.
In an implicit way everyone uses a checklist as you do. Weaker (or more unexperienced) players, maybe need to mentally go through all the points in order, whereas stronger (or more experienced) players are able to do so automatically at every move. When my opponent moves a piece, I automatically check which of my squares and pieces are now attacked and create a threat to me. Similarly I also check which squares are now less defended and if that opens an opportunity for me. Similarly I check for king's safety, tactical motives and other opportunities. This is however not explicitly checking if there is, for example, a fork attack pending, but more the realization that two pieces are in a position where a fork attack might happen (regardless if it is currently possible or not).
Nevertheless I sometimes take some time to evaluate the whole situation on the board. This includes tactical motives but rather focuses on strengths and weaknesses of my position and my opponent position, ideally leading to a plan that can later be executed. Here I often fall back to a checklist, looking for common features such as weak pawns, inactive pieces or open lines and diagonals.
As advice for you, there is no shame in trying to avoid blunders. If you have to look explicitly for tactical motives, then do it. With time you will get used to the most common patterns and find them more easily and quickly. Maybe you will also be able to find these patterns "by heart" (that is without having to look for them). But even if not, your opponent will never know and your strength doesn't come from the way that you look for blunders, but how effectively you can identify them.