Does saturated fat result in crispier fried food?
Having looked into this a bit, I have some hints for an answer. I'm still not convinced I understand the full mechanism, so any feedback is welcome here.
I think some of my sources might be paywalled; apologies for anyone unable to access them.

Reading a few studies on the effects of different variables on the outcome of deep frying (specifically potato chips), I found the following information.

Oils higher in saturated fats are more 'stable', that is, they show less change in chemical composition when heated (3).
Hydrogenated vegetable oil in one test (1) provided the crispiest results when compared to refined soya bean and groundnut (peanut) oils. The difference got bigger with increasing temperature, suggesting the stability of the more saturated oils has some effect.
This last point is corroborated by (2), who showed that the difference between different oils is biggest at higher temperatures.

The results from (1) in particular support the idea that more saturated oils result in crispier foods. The question why still remains unanswered, however.

Side notes:

While (1) found that the hydrogenated vegetable oil produced the crispiest results, people on average preferred chips fried in peanut oil.


Sources:

M. Rani and G.S. Chauchan, Effect of intermittent frying and frying medium on ihe quality of potato chips; Food Chemistry, 51 (1995), pp. 614-617.
A. Kita , G. Lisińska and G. Gołubowska, The effects of oils and frying temperatures on the texture and fat content of potato crisps; Food Chemistry
Volume 102, Issue 1, 2007, Pages 1-5.
E. Choe AND D.B. Min, Chemistry of deep‐fat frying oils. Journal of food science 72.5 (2007): R77-R86.