I seem to recall that there was a discussion about the lack of a video for 'Back in My Life Again' a few years ago. I can't remember what the upshot was, but I'd suspect it had something to do with which singles were being picked for markets outside the US.I don’t think Richard (or Karen for that matter) has ever gone on record about his thoughts on the chart performance of their singles from the album. It’s almost as if they kept putting out song after song, hoping the next one would do something and the chart listings got progressively worse. Putting Beechwood out in March 1982 was just a waste of time and money. The album was nearly a year old by that point and they weren’t even a functioning unit able to promote it. I also wonder why no video was made for (Want You) Back In My Life Again when all the other singles had one.
I'd have to double-check, but I think 'Back in My Life Again' wasn't released as a single in many other markets - maybe only Brazil? - whereas 'Touch Me', 'Those Good Old Dreams' and 'Beechwood' all saw release in a number of countries. As such, they were given the video treatment so that these could be used in those various markets.
This does of course rasie the question of why 'Back in My Life Again' was largely overlooked as a single outside the US when it was supposedly one of the more forward-looking tracks with single potential.