One thing I've done mostly as a side hobby is downloaded the audio files for the loops directly from the NASA Audio Collection on Archive.org, ran a noise removal algorithm in GoldWave to get rid of the warbling noise in the background produced by the tapes, and then a silence removal to get rid of the gaps. The result is a relatively seamless track that can be listened to through a multimedia player like VLC or AIMP. I could produce a step-by-step guide for doing this if need be.
Of course, this is not the same thing as skipping the files directly through the Apolloinrealtime.org website, but I can imagine that being a general pain to program in.