The airport has a legal obligation to ensure de-icing contractors are familiar with both the rules and specifics for individual types, having said that, commercial pressures, boredom and being human means mistakes are made. but the ultimate responsibility for ensuring the aircraft is fit to fly is with the Captain.
Cabin crew are usually briefed in cold weather ops and in fact encouraged to look out of the windows, the wing inspection lights are not just ornaments.
Flight deck to ground ops and cabin communications should be sufficient to cover most scenarios, of course you cannot plan for them squirting guilds where they are not meant too. e.g 737 horizontal stabiliser tracks.
It is not practical for the Flight crew to physically inspect post de-icing, unless it was done on the ramp before boarding and there are no HOT issues, having said that it is again usually not practical as pax marching through dripping fluid and contaminating the interior and steps is a bigger problem.
De-icing with engines running, or one running is not uncommon is packs are off and the crew make an had about the blasting going on outside,
This event merely shows the system "works" and cabin crew suggestions should be encouraged, of course, we've all had the "know it all" pax in the cabin, "difficult decisions are the privilege of rank"