"It is quite common here (in the US) for touring Irish and Irish-American groups to actively encourage clapping along." -- I think the clapping probably originates with "commercial" groups, whether in the US or in UK/Eire, who try to involve the audience by getting them to clap along with fixed beat songs. Audiences then think that this is a way of showing appreciation and involvement.

"A solution would be to tell them all to hold hands for peace" .. "Just announce that it is too difficult for you to play when people are clapping" – It's often difficult to make "announcements" if you're in a session as opposed to performing on a "stage".

"I personally like the Scottish notion that rhythm should be very strict" -- The problem is the clapping often isn't!

My own experience of sessions and singarounds (in the UK, not Eire) is that musicians and singers are not there primarily to entertain an audience but are there because they love the playing and singing. However, when you're in a crowded pub and there's a group who feel that they're an audience, they don't understand that you're not there just to entertain them, so they continue to act like an (appreciative!) audience.

So – I think it's a misunderstanding caused by a clash of cultures. Somewhat irritating but pubs are, as the name suggests, "public" places.