For a more data driven approach, here's a google trends graph for the popularity of searches for "climbing gym", "indoor climbing", "climbing center", "climbing centre", and "climbing wall": google trends
Nothing too illuminating, outside of an interesting biannual cycle (peaking in the middle of the winter and the middle of the summer). Interestingly, there also seems to be a taper in interest from 2004--~2008 and remaining relatively constant over the last decade, which also jibes with looking at trends for just "climbing". Also of interest is the regional variation in preferred search term.
Going back further, we can look at google's nGrams analytics, which looks at the prevalence of a term appearing in books. For the same search terms, we get the following trends: google nGram
Of interest are the large spikes c.1900, 1930, and 1950 for the term "climbing centre." Clicking through to some of the relevant texts, this appears to be used in classic British mountaineering literature as a term for what we might now refer to as "climbing areas" or "climbing destinations." If we ignore these spikes as spurious data, the remaining trends seem to jibe quite well with the general climbing narrative: a rare few gyms appearing in the postwar era, but with a major explosion beginning in the late 80s to a nadir c.2001. That there is a distinct peak followed by a taper in appearances since the new millenium seems to confirm the initial decrease in interest in the above google trends chart.
Most of the truly old appearances for "climbing wall" seem to be for gymnastic training, for example the 1867 Handbook for Athletics and Gymnastics contains the following image, predating the lattice board by 150 years. Note that it suggests "campusing" as an exercise a full century before Wolfgang Güllich.