Mumsnet is so bonkers for parenting advice now, I swear it used to be sensible. Maybe it's just AIBU.
Of course there's a middle ground between strapping him weirdly into a buggy and sitting him there to watch you as he screams the place down while everyone awkwardly drinks tea and tries to pretend they haven't noticed while thinking "God what in the world is she doing??", vs being in thrall to your tiny dictator's every whim 
Don't worry OP I'm sure you're doing absolutely fine.
If the routine has recently changed, then he's just doing what he's learned from previous times - toddlers are creatures of habit. And depending on age and personality you'll have varying success with distraction, explanations etc, because that story is set in his mind and it will take time to adjust it.
If the place is secure and toddler-proof, which is generally what I'd expect from a stay and play session, then I'd probably just let him wander around and hang out by the door/gate if he wants to while you drink your tea. After a few sessions he'll get the message and go off and play instead. If he is shouting "Mummy, mummy, come/time go to" (or whatever) then you could just call back to him "Mummy's just drinking her tea sweetheart, it's not home time yet." Then if he became noisy to the point of annoying others, see below.
If it's not secure and he can leave by himself and this is a safety issue, then it depends on age. Below around 2.5 you will have more success with distraction/redirection - go to him and explain it's not home time yet, we're having biscuits/tea, or take him to a toy and see if you can get him interested in it. This will probably be just as demanding of your time/attention as chasing him, so it's a long game but after 2-3 weeks of this usually they'll start to get the new routine and then you'll be freer to chat if you want to.
Over about 2.5 you can try this but combined with explanations. Try some before the session "Normally we go home after the bye bye song, don't we? But next time, we're going to stay for a biscuit. Would you like to have a biscuit after bye bye song? Bye bye song, then biscuit, then home." You can even do a pretend Moo Music session with some teddies and model the tea-and-biscuits part there, if you want to. Remind him on the way there. Remind him again when they are setting up the goodbye song "Oh look DS, it's time for bye bye song, nearly biscuit time!" He will probably still go to leave again but you can then remind him again "I know we normally go home, but today we're having biscuit time!" They need SO much repetition of a new concept to override an old concept which has been set in place by repetition. That's the reason for repeating yourself multiple times in this instance because it helps shortcut that rather than it being a week between each instance of the new pattern.