I used Eurostar in the summer and the capacity problems seemed to be largely passport control but only half the booths were manned. As a result trains were being delayed waiting for passengers to clear the border controls. Eurostar were also printing on the tickets that they recommend you arrive 90 minutes before departure but if you did refused to let you even join the queue more than an hour beforehand. It seemed very poorly managed but I suspect a lot of the problems could be avoided if passport booths were fully manned.
I used it again just over a week ago again with two departures fairly close together and this time all the passport booths were manned and whilst the trains looked pretty full it was working much better with trains leaving on time. Then the only problem was the badly overcrowded departure lounge but I feel that could also be improved by allowing passengers on board more than 10 minutes before departure as many would then leave the lounge.
Airports don't seem to be struggling to anything like the same extent. I know early summer was a problem but since then most airports don't seem to be taking longer now than before Brexit. Heathrow and Schiphol I think did have to cut some flights due to staff shortage but that seems to have settled down now and in the case of Schiphol at least Brexit won't have been the cause so I am surprised it is having such an impact at St Pancras.
I don't think Eurostar are really helping themselves though with Ashford and Ebbsfleet both closed forcing all passengers into London then complaining that the terminal there cannot cope.
So a lot of the problems are due to bad management by Eurostar at St Pancras, and staff shortages by UK and/or French border control.
It is pointless to recommend that people arrive 90 minutes before departure if check is not open, and people are not allowed to join the queue. This just leads to lots of people hanging around the station, and a rush of people when check in for a service opens.
My experience with Eurostar over the years is that they can be very variable about when they open check in. Sometimes it can be as much as 90 minutes before departure, and other times it is as little as 45 minutes.
If Brexit is increasing processing time, then you need to make sure that all passport booths are manned, and allow sufficient time before departure to process everyone. I have used Eurostar twice since Brexit, and both times there was no noticeable increase in processing time.
Spacing out of train departures would also help, and if this is not possible due to pathing issues, if there are a group of departures close together, then check in should open 2 hours beforehand to even out the flow of passengers through the system.
There is no reason why UK passport holders shouldn't be able to use the e-gates at St Pancras, even before the ETIAS scheme starts, especially considering that citizens of EU member states are allowed to use the e-gates at Heathrow on arrival into the UK.