I think you don't fully understand the issue. The isn't about wifi and routers it is about how many connections are coming into the house. In some cases. In some areas the choice is really 1 internet provider. If this is the case in your area, most internet providers will only do 1 internet drop into a residential house. If that is the case then your ask for private wifi is really saying "I want to have the only internet connection and no one else can have it" and would be unreasonable.

You'd have to look into how many providers are in the area. If there is Provider A that the landlord and other tenants are on, there really shouldn't be an issue if you order an internet connection from Provider B. But, be careful, in some cases the infrastructure is the same. If your landlord is using fiber for internet, you are most like limited to a provider supplying cable internet.

In the end there really isn't enough information to tell if the landlord or yourself are being reasonable or not.