You’d be surprised that actually most people, even if taught, are unable in reality to accurately find a pulse on a living patient! It’s not as easy as one might seem, and in an event where someone isn’t breathing, the adrenalin rushing through you will make it even harder and less likely you’ll actually be able to determine whether or not they have a pulse.
It also boils down to the fact that if a patient does ‘die’, the first process to stop is breathing (hence why we check breaths), the heart continues to pump, though weakly for a short period of time afterwards, though this does not mean the patient is alive, they are not. This means that if you did find a pulse it does not mean the person is alive, hence checking to see if the patient is breathing is a much more reliable and quick life saving method.