well illness is allways a multi factor thing.
You are all right about the distraction and the increased mood having therapudic effects.
As far as stimulating you body via adrenaline/stress reponse systems, if it does that could be positive (like laughter is) or negative, if your body is preparing itself to flee or fight the saber toothed tiger (or Britons you are playing against), then it pulls energy and resources away from your imune system (this is a bit simplified, but for illness recovery this statement hold VERY true). Thus, stress may cause increased blood flow, but may take energy away from your bodies immune functioning.
However, if it causes enjoyment, and is not causing any stress response, then it could be a good form of distraction and still allow you to rest in the process.
Also there are factors like, the immune (and stress response) sytem when kicked into high gear can take energy away from your stomach, as a result, eating "heavy" solid foods could cause (mild, tmporary) gastro intestinal problems. By playing games all day, you may be cutting back on your food intake (i.e. don't eat while you play), and thus lead to less discomfort that way.
Also, drinking fluids and consuming cough drops (and a few pain killers) can make that soar throat and ear infection that was killing you in the morning, almost unnoiceable. It is very common (I think [the doctor implied to me once]) to feel your worst first thing in the morning and feel better as the day goes on, then bad again the next morning. So by playing video games, it may only be a misatribution of these common effects taking place.I would be careful if you're missing school work or skipping classes b/c u r sick yet playing games, the rewards could cause you to unconsciously engage in negative health behaviors that could make you more likely to be sick, and thus rewards you by allowing you to replace unfun activities with rewarding video games.
All in all, I say play if it makes life more enjoyable. Because being sick can suck, so make the best of it.