Lets face it, Youtube won't last forever,
I see "Let's face it - X won't last forever" along with "all good things must come to an end" and "X will be replaced eventually" continuously. YouTube won't last forever but it will last a very long time. YouTube is now the primary search engine for videos and is partnered and supported by Google, the primary search engine overall. YouTube and Google combined will last a very long time because of their absolute dominance in the knowledge and media departments - no consumer website can give up on either of them. That's why I don't think that's a good argument.
In terms of gaming content, it won't fade until either A) People are bored of games (not likely), B) People are bored of gaming content (unlikely unless YouTube changes something) or C) People stop using YouTube (extremely unlikely in the short term). I think gaming content production is a stable job - the instability people complain about in terms of YouTubing is either YouTube's unpredictability or the person's LP audience losing interest.
What will they do long term? Will they be playing video games and uploading them when they're 60?
Why not? The problem is that there have been no cases of YouTubers retiring when they're 60. I think about it this way - T.V. presenters in general are similar in job description to a full-time YouTuber (editing and rendering replaced by paperwork). They don't need to retire at 60 if they enjoy what they're doing. Likewise, if a YouTuber has a drive to continue to play/ upload games, there's no age limit to uploading YouTube videos.
Is it better to do lets plays and if you get paid for them do it part time while keeping your full time job?
Depends. Full time job, not career vs YouTube? I'd say YouTube. Full time job as part as a career vs YouTube? Bit more tricky, up to the circumstances.