These are not quite the same thing, although I think it can be kind of tricky because
していたい has no obvious direct parallel in English. It expresses a desire to be in a state, although if you express that desire while in said state the distinction obviously becomes very fuzzy.
Well, the very obvious one is that you can
していたい a state you aren't currently in. Here is a question about
people who constantly want to be busy. We can reduce this to 忙しくしていたい, which is just
want to be busy. You can say this if you aren't currently busy, whereas constructions with
続ける are not appropriate in that situation. Of course,
忙しくしたい also just comes out as
want to be busy in English, but this isn't what you are asking about here - I'll just say that between
していたい the latter has more emphasis on state over time.
The last thing I would add, although I think we are venturing into fairly subjective territory, is that I think there are cases where
続ける implies more agency than
ている. ずっとこうしていたい is the Japanese equivalent of English's
I want to be like this forever which you might expect to hear in a romance movie, but I think
ずっとこうし続けたい would end up sounding weird here because there's no explicit action that you want to continue.
I think you have answered your own question here, in the sense that
ていさせる sounds pretty weird to me. I would not say this, although perhaps a native speaker can come up with a situation in which it might be appropriate.
Unfortunately, as with most things in language, the answer is simply "because people don't say that".