I am not altogether sure if the issue is whether the DM candidate can swim (although that is a legitimate concern in general

). Rather, because the swims are timed, and you need to move somewhat expeditiously in order to get a particular time, that earns specific points toward the required 15, some candidates struggle.
In the case that boulderjohn referenced, my understanding of the event is that it wasn't so much a matter of whether the DM could swim, or even swim fast. Rather, the DM apparently entered the water completely unprepared to assist a diver in distress, and probably did a (very) poor job when he reached the unfortunate victim. That is a matter of DM training, preparedness, competence, and confidence, not ability to swim.
I recall my DM class, and feeling the same stress as the OP. I (thought I) was a good swimmer - I grew up on the VA coast, swam a lot, etc. But, I never swam for time, and I therefore didn't have an optimal time technique. I just swam, I didn't worry about pacing myself for a distance. And, when I did the 15 minute survival float the first time, I didn't get a 5 - because I thought I was supposed to hold not just my hands above water, but my arms up in the air.

When I realized that the performance requirements say only that the hands need to be above water, I did just fine. The first time I did the Tired Diver Tow, I started out like the often cited 'house afire' and ran out of gas 3/4 of the way through. What would have been a 5 if I could have completed the tow at the same pace I started with, turned into a 2 as I crossed the line like a very slow (and winded) land turtle.
To the OP: physically fit DMCs, who can certainly swim but aren't used to swimming timed laps, or swimming for time, may struggle at first to meet the point requirements. You are not alone. It is also not a 'wash out' requirement, that you can only try once. You can try several times to 'get it right'.
For the 15 minute water tread -
relax, minimize your motion, use your inherent buoyancy to stay afloat. As long as your nose is above water, you can breathe.

Lie on your back if you want to. When you hit 12 minutes, put your HANDS (only) above the surface, not your whole arm, not even just your forearm. For the timed swims, practice in a pool to see what the pace
feels like that will get you a 4 or 5. Time yourself by lap. Manage your early pace, to save energy for the last 1-2 laps. If you get a 5 on the tread, a 4 on one of the swims, and two 3's, you have your 15 points, before even moving to the 5th exercise.
Now, a personal anecdote: as a diver I swim, underwater, A LOT. But, I also frog kick most of the time, I swim s-l-o-w-l-y, to enjoy myself, to conserve gas, to save energy. Oh, and by the way, when I swim underwater, I wear big things on my feet, called fins! And, I NEVER use my hands to swim, it is a matter of pride to control my propulsion, and direction, with only my legs. Several years ago, I decided to do a sprint triathalon with two of my adult children - a swim, a bicycle ride, a run. My goal - finish the course, not worry about time. I knew my running time would be awful - I just don't run anymore. But, I also figured my bike time would be very good (it was), and my swim time would be respectable. We were batched in groups of 20 participants for the swim (the first event), and the groups were started 5 minutes apart. I was pumped. My group got in the water and started swimming. And, I felt like I was going nowhere fast. I was just not getting much propulsion, from my arms which I actively DO NOT USE while diving, or from my feet, which seemed to be useless without those big fin things. By the time I finished, I was not only the last one in my group to reach the other side of the sound, I had been passed by everyone in the group immediately behind me, most of the swimmers in the group two starts behind me, and even one person in a group 3 starts (15 minutes) behind me!
As a scuba Instructor, I can swim, I can swim pretty well, better than most. But, I sure won't set any speed records.