Maybe the author thought it was so obvious that you could get some lower bounds that way, that it didn't seem worth mentioning! :) Wikipedia has a list, where I presume the lower bounds are mostly demonstrated constructively. Upper bounds must be demonstrated non-constructively, so I presume computers don't really help there.
https://en.wikipedia.org/wiki/Kissing_number#Some_known_boun...
Even in dimension 5, the kissing number is apparently known only as "42 plus or minus 2."