"Acceptee" Is Often Ambiguous
The Offeror and Offeree construction, like Lessor and Lessee, always comes in a pair. An act and a response to the act. But, following that logic, the "Acceptor" should be the person who accepts an offer to enter into a contract, and an acceptee should be the person who made the offer, since they receive the acceptance.
According to the OED at the link in the OP, the word "Acceptee" has two senses which mean basically opposite things.
One sense is synonymous with Acceptor in the quoted language in the OP, and is the person who accepts an offer, but it is confusing because the construction is not parallel to Acceptor.
The second sense in the OED is one in which Acceptor is an entity that admits someone to membership (e.g. a college) and Acceptee is the person who is admitted to membership (e.g. a prospective student who decides to attend a college). But this sense isn't naturally applicable to contracts and involves something that isn't litigated.
So, the meaning of the word "Acceptee" is often ambiguous in the context of offers and acceptances of proposals to enter into contracts.
Early Uses Of Terminology Become Entrenched In Common Law Systems
Beyond this, there isn't any profound reason.
Neither "Acceptee", nor any of the other words in question (like Offeror, Offeree, and Acceptor), are used with any regularity in statutes.
All of these terms are used principally in the context of case law interpretations of common law contract rights.
Common law terminology is prone to strong "founder effects". Once a term is used once, subsequent courts citing past precedents like to quote exactly from previous court rulings.
Often, there is more than one word which would be appropriate to describe a participant in a contract formation process, as you point out, but a court will generally use only one in any given court opinion for clarity's sake. Calling the same person an Offeree in one paragraph of a court opinion, and an Acceptee in another, would frequently be confusing.
In part for the substantive reasons, but also just as a matter of random happenstance, "Offeree" came to be used in some of the early opinions rather than Acceptee, and everyone writing court opinions and making legal arguments who followed tended to imitate that early word choice so as to invoke the legal authority of the early precedents.
So, one term "went viral" in the common law opinions that cited the early decisions, and another one which would have been equally suitable fell into disuse in legal writing.