There is one sense in which a landlord may have to "let" a tenant stay, which is that they cannot physically evict a holdover tenant themselves. The landlord has to file suit against the tenant, and if they win the case the sheriff will eventually remove them. In this scenario we are talking about a matter of days.
But if the landlord permits them to stay over, he cannot also sue to immediately evict the tenant. The language of the lease could itself grant permission to change from a yearly lease to a month-to-month lease. Or, accepting another months rent would constitute giving permission to remain. In that case, the landlord would have shot himself in the foot, leaving himself little legal recourse: he would be in breach of contract w.r.t. your lease agreement. You have a fourth option, namely to stay somewhere else for whatever period of time it takes to make the unit available to you – probably the landlord will not be happy with that option (this is an option that your lawyer would mention to the landlord). A fifth option, which I presume the landlord also didn't mention, is that you can petition the court to order the landlord to comply with the terms of the lease, or to compensate you (more than just the refund).
The basic principle is that your lease is a contract, and both parties must abide by the terms of the contract unless there is mutual agreement to change the terms of the contract. In case of breach of contract, suing the other party is an option. In some cases the court might order a party to comply with the terms of the lease, or, it might order the damaged party to be compensated. As the damaged party, you have a duty to mitigate your losses. That could involve accepting an alternative unit – as long as the alternative is a reasonable substitute. People pay extra money for a good view or for convenience, there could be a case that e.g. you deserve some additional compensation.
These are basically issues of negotiation, which your attorney would conduct on your behalf. The question of whether the landlord has some legal obligation to another party is irrelevant to you: what matters is that he has an obligation to you, and it appears that he intends to not honor that obligation. Which is why you need to get a lawyer to initiate proceedings against the landlord, if you cannot come to a satisfactory agreement.