On-train border controls have some advantages IMHO (no need for specific infrastructure is the big one and the fact that nobody can exit/escape) although the main worry would be the inefficiency of such a thing (how many BO would you need to scan 900+ passengers twice in less than an hour (for a Brussels/Paris service) without any eGate or other systems. On the other side, apart from stopping at ebbsfleet, that would be even harder
Having to force disembark passengers at Lille would probably make a lot of headache for the French (unruly passengers, risk of cleared people sneaking out...), for the UK (if you need to conduct a interview) and for Eurostar (having to make sure everyone that need to disembark have disembarked and probably having to deal with sending them back from where they came from could cause a lot of delays)
Having to force disembark passengers at Lille would probably make a lot of headache for the French (unruly passengers, risk of cleared people sneaking out...), for the UK (if you need to conduct a interview) and for Eurostar (having to make sure everyone that need to disembark have disembarked and probably having to deal with sending them back from where they came from could cause a lot of delays)