Earlier quoted context omitted.
Salt water is buoyant. If you know how to float, it doesn’t take much energy.
Some people simply cannot float. Me included. I've had 3 different instructors try to tell and show me that everyone can float. I can't. I have 5 swimming certificates, I'm fit, but I think I just don't have enough body fat? Even in the Red Sea (if it's called that in English) I can't float. Not for more than 5 seconds.
When I started training in freediving (apnea), I learned how to take much fuller breaths. I can almost guarantee that if you haven't done that kind of training, strengthened your diaphragm and learned 3-zone breathing, you aren't taking in nearly as much air as you could.
Our bodies are mostly water and the amount of mass that we have that's denser than water isn't that much denser. But water is a lot denser than air and the more you can add air volume, the more buoyant you become. Learning to float is a matter of learning to maximize your volume. Some density numbers from the internet...
Salt water: 1.03g/ml Fresh water: 1g/ml Muscle: 1.06g/ml Bone: 1.1g/ml (approximate...couldn't find a good source for this) Fat: 0.9g/ml Air: 0.0023g/ml
The numbers seem to back up your instructors. As you can see, all these values are fairly close except for air, so adding more air to your volume can dramatically change your buoyancy. Wikipedia lists the average adult male lung capacity at 6L, so a 5L breath should be achievable for most. In salt water, that 5L breath will add more than 5kg of buoyancy, which is enough to float around 400 lbs of muscle or 176 lbs of bone. In fresh water, that same 5L breath will add a bit less, enough to cancel out 193 lbs of muscle or 121 lbs of bone. There's no way that I can know your body composition, but those numbers are very substantial. The more likely is that you're just not getting anywhere near 5L of air in your lungs.
If you're ever into you're trying it again, look up some stretches to open up the lungs. Then practice breathing shallowly with nearly-full lungs. If you're able to do that, you'll probably be able to float laying on your back, where your head is mostly submerged...remember, every part of your body above the surface counts against your buoyancy, for the same reason that air in the lungs matters so much.