"Also Tuck sounds more like a surname (they were just beginning to gel in England then)"
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I just stepped over to my unabridged dictionary and looked up "tuck." To my surprise, the oldest meanings of it are quite negative. It comes from the Anglo-Saxon "tucian" to ill-treat. Early meanings included to punish, torment, torture, scold and upbraid.

It is akin to the words "jerk" and "tug."

We can safely conclude that Friar Tuck was not a saintly guy.
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As for the BBC not wanting to deal with him (supposing this is not merely a rumor), don't you think that current pop culture is sadly lacking in humor? When did you last hear a funny song? Name a famous comedian of today.

Often what passes for humor is mere meanness or vulgarity.

I think that the trouble with humor is that it's hard to produce regularly and on demand, and so the networks have ditched it.