I am sure we could have Border checks done on the train, if the train company was prepared to pay for it. But this would render the service uneconomic. Currently, on the Paris service, there are about 4 members of the UKBA stationed at Paris Nord to check passports of one, sometimes two, trains per hour. Two 8 hour shifts, x 4 staff members = 8 staff members to check about 16 departures per day. Any passengers without correct travel documents are refused travel and remain in France. Contrast this with the costs of checking documents in the train: each train journey takes about 2.5hrs, so one round trip of UKBA officials (travelling passenger on the outward and say an hours layover in case of delays) would occupy a complete shift. Would 4 staff members be sufficient to check passports and deal with, and secure, people [ how many, could be as many as 15 on a single trip?] who do not have the correct documents. Just say 4 is sufficient per train, x 16 trains = 64 staff members. Plus staff at St. Pancras to detain those without correct documents and ensure they are kept secure and returned. Plus deal with the problems of those who destroy their documents and are not then accepted back into France. Airlines have to pay a large fine (as well as accept responsibility for return) of anyone who arrives in UK without correct documents, hence their checks of the documents on check in. Airlines do not usually mix Domestic and International passengers on the same plane, and in those rare circumstances where this may happen, seat occupation is much more rigidly applied. The railway industry would be subjected to similar rules.
Whilst a short local may require fewer UKBA staff due to the shorter journey (say 1hr Calais Frethun - Ashford) meaning each crew could do 2 or possibly 3 round trips in a shift, detention facilities would also be required at Ashford, and a holding cell provided on the train. (I believe Eurostar trains have this facility?). I doubt a short local train would be seen as an efficient use of a tunnel path, and would be very expensive on an access charge per seat basis. Additionally there are different, more onerous, rules regarding subsidisation of International train services, one of the reasons for their gradual reduction all over the EU.
Whatever is thought about Schengen, there are many deep issues and it looks unlikely that the UK will join soon. (What Politician is going to vote for this and run the risk of wrath over immigration issues? - certainly not until there is a real economic argument for it; cross channel local trains are not going to do this!). It looks like that until Britain joins Schengen, local cross-channel trains are unlikely.