Do some incorrect or imprecise terms stick just because English language hasn’t better options?

Never, never, ever underestimate the power of inertia.

In the instance you cite, of sex addiction vs compulsivity: the distinction is itself fairly new, and the use of the description to describe the patient has not yet stabilised, because the notion of compulsion as a medical condition has not been pervasive. So there’s a huge amount of inertia behind addiction, and an even huger amount of inertia because there hasn’t been until now a term for “one suffering from a compulsion”, to match “addict = one suffering from an addiction”.

If we went to Latin, we would use the past passive participle, find it to be compulsus (cf. addictus), and say that the person is a *compulse. But that hasn’t happened in English with any of the -pulse/-pel verbs. Not least because pulse and impulse as nouns are abstractions.

Since a sufferer of compulsion is grammatically one who is compelled, we could use *compelee. But compel and compulsion have actually diverged—compel is not used in the psychological context.

So, by accident, we don’t have a straightforward derivative word to describe such a patient. What to do, OP, what to do…

… actually this has already been solved. compulsive can be used as a noun to describe someone who exhibits a compulsive disorder: a sex compulsive. This is also something that English does with adjectives; cf. captive prisoner > captive. It sounds odd to us, because compulsive disorder describes the compeller and not the compellee; but it’s better than the alternatives, and it’s already in use.

The grammatical strangeness may slow down the take-up of compulsive; but if there is a compelling (ha!) case for a single word to be used for sufferers of compulsion, it will be taken up anyway.