Are you sure it is a CRT and not a projection TV?
Most definitely a projection tv. CRT's do not have a "bulb" per say because they are "Cathode Ray Tubes". it is not something that can be taken out and replaced like a light bulb where as in a projection tv it can. in the original post it mentioned that it has blown out twice. if it were a CRT then the only way to replace it would have been scrapping the entirety of the internals of the tv and replacing it.
my in-laws had a projection tv up to 2 years ago when the bulb blew on it and they could no longer get any replacement bulbs. it would turn on but would have no display because there was no back light. to me this sounds to be exactly the same issue. check the model number online and see if you cant find a replacement bulb for it. (word of warning though they typically cost around $80)
as for the original question no. playing video games on a tv will not cause them to fail. it is exactly the same thing as running your tv services through it (Mediacom, dish, direct tv, ETC) the device you use processes all the information so there is absolutely no additional load on the tv as there would be if you were watching static on it.
also another thing to keep in mind is that they physically stopped making those bulbs for the projection tv's 10 years or so ago so the bulbs they are buying, even if they are brand new, are at least that old and as such have a much smaller working life.