If your site doesn't need to be searchable, does it NEED to go responsive?

If I have a client who has a couple of very small unresponsive sites, and he says he doesn’t need them to be searchable on google, he only refers clients directly to them, so does he still need a new responsive site? To him it doesn’t matter if Google ranks them lower, but I’m trying to pursuade him that it’s still a good idea, or am I wasting my time?

Surely it would still be useful if clients could view them on mobiles? It’s not as though it’s hard to make sites responsive these days.

Hi site is still visible on a mobile of course, even though it’s static, he doesn’t necessarily want to pay for the privilege of a responsive site unless it’s absolutely essential.

I’m finding myself having more discussions with clients about the “responsive” theme choices I pick.
They seem to like the “static” theme which is just smaller on the mobile device. They can still easily navigate by pinching in and out. I then start looking at older themes. If “static” themes (or the option) are dead then it’s going to be an interesting future for me finding new themes for clients that prefer one look on all devices.

I don’t agree, for me it takes quite a lot of work turning an existing site into a responsive one and it’s hard to charge enough considering the time that goes into it because it’s already existing.

Yes I agree, and any old sites I turn responsive are probably going to look different to the original which they loved anyway.