Whenever I see the continuous time Fourier transform being introduced in books about Fourier analysis, it is as an extension of the continuous time Fourier series. The issue I have with that is that I always get lost in the steps taken; There are simply too many substitutions for me to keep track of.
Is there a proof of the relationship between the CTFT and the ICTFT that does not rely on Fourier series?
If you are referring to the Fourier inversion theorem, then yes, there are multiple proofs that do not rely on Fourier series. One proof uses the fact that Gaussians are eigenfunctions of the Fourier transform. (In fact, this is the proof outlined at the end of the Wikipedia article.) Another proof uses complex analysis and involves contour integration along a rectangular contour. A proof along the lines of the former can be found in Stein & Shakarchi's textbook Fourier Analysis: an Introduction, while the complex-analytic proof can be found in Complex Analysis by the same authors, among numerous other sources.