Are linear regression models with non linear basis functions used in practice?
Yep, that's a thing. It's called a "Generalized additive model (GAM)":

https://en.wikipedia.org/wiki/Generalized_additive_model

You may also be interested in "multivariate adaptive regression splines (MARS)":

https://en.wikipedia.org/wiki/Multivariate_adaptive_regression_splines

EDIT: Regarding demonstrating that these are actually used, I'm not sure what you're looking for. I've never seen a survey to try and gauge the popularity of specific models, and I'm not sure how useful such a thing would be. I could speak from my subjective experience, but I don't consider that particularly meaningful either. 

If you just want examples:


Hits for GAM models on PubMed - 2114 hits
Hits for MARS models on Pubmed - 190 hits


For context:


Hits for Random Forest models on PubMed - 5604
Hits for Deep Learning on PubMed - 4435 hits
Hits for Neural Network on PubMed - 57170 hits


Take these counts with a solid dose of salt: in my experience random forests are in much wider use in practice than neural networks. But like I said earlier, you shouldn't really trust my subjective experience either since that's just anecdotal.