The purpose for checking on the French side was probably in part to make sure the Chunnel was protected from the greater potential spectrum of threats that might come from the continent, what with its open borders.
I would consider it likely that these were consolidated on the French end both ways as a cost savings measure.
If that was the case the Brits should take a stand on the basis of continued Chunnel security and offer to cooperate with checks at BOTH ends, where inbound, and make this a cooperative measure at both ends (iow: neither is relying fully on the other yet neither loses face).
And they should further also announce a cessation of ANY refugee traffic of any sort via the Chunnel route. The Chunnel is ONLY for daily, normal traffic and no other. Then the Camp awaiting transit to England near Calais has no compelling reason to exist there (the old ferry services having long since ceased for the most part) and the French gain a needed pretext to clean the concentrated nest out and break it up, sending its denizens elsewhere where they will lack numbers presently held.
Too complicated. Just deport them to their home countries and shoot them if they resist.
There are already checks at both ends. Under the Le Touquet agreement, French inbound customs/border checks are at Dover (and St Pancras London for the Eurostar trains.) That’s why the French are officially rather cautious about tearing it up. Incidentally the ferry services are still alive and flourishing.