I recall that it was called "Betta-builder", not Brickplayer. However, from Googling, you seem to be right, Betta-Builder being something described as a cheaper version of Lego. I wonder if there was some brand-name selling that went on.
The alternative building toy then was Bayko, much quicker but far less realistic. I recently sold a complete Bayko set in its original box in its original with instructions (I was a careful child) on ebay for a good price. I only ever saw a Brickplayer house built once (in a friend's house), and I think after someone had built one they did not dismantle it again!
Lego took over from both. Lego was originally seen as a building toy - ie for miniature brick-laying for which it is very good. I think it is a great pity however that it expanded into "engineering" with wheels, gears and hinges, largely displacing Meccano. I am an engineer and Meccano is a much more realistic representation of engineering; in fact it is sometimes used in engineering design offices for demonstration of priciples with rough scaled down "prototypes".
Perhaps that has something to do with the demise of the engineering industry in the UK. I acquired what you might call an instinct or feel for the priciples of structural and mechanical engineering with my Meccano long before I was taught them formally. I have helped my children with Lego and I don't think I would have acquired it that way.