I think we need a permanent answer to this question, as it comes up about every three months!
1. It is horrendously expensive in terms of staff resources. The team required to process a train with 800 passengers, with ample capacity to deal with issues, would be very large, in order to complete a train from Brussels to Lille/Paris to Calais, each with Team Leader(s) and Security
2. Passports now have to be digitally checked with on line databases, the connection to which are not reliable on a fast moving train.
3. Every train would have to stop at Lille/Calais, team disembark with any undocumented persons and then travel back passenger to the starting point. Each team would be fortunate to manage two trains per shift [ if the times were right] - on longer journeys the team could be smaller but only manage one train (see post #44)
4. There would need to be a private security presence at Calais/Lille to deal with undocumented passengers, escorting them out of the secure area, and dealing with those reluctant to alight. The French authorities would doubtless not be keen on assisting - after all it is not their problem. The potential for train delay, and resultant cost and passenger inconvenience would be enormous.
All of this extra expense would have to be covered by increases in the ticket prices.
It is obvious why the system is as currently. Yes, E* could adopt the airline model, but as explained, I am not sure how this would improve the situation - check in, secure areas etc would still be required, and St Pancras needing reconstruction and arriving passengers delayed.
I know passport checks used to be done on board trains in European countries in the past, but that was in a different era and is simply not appropriate now. How many places around the whole world is passport checking done on board a moving train now ? Very few, and probably none on a similar situation to E*.