Having used Eurostar both pre and post Brexit I do not believe the need to stamp passports is the problem, the time spent at the French Border police window has only increased by a few seconds. I think the issue is lack of staff, and the fact that St Pancras can only process about 1800 per hour. Closing Ebbsfleet has to have increased pressure at St Pancras, Ashford less so. I do feel blaming it on Brexit is at least in part politics, I think the reality is even without Brexit St Pancras would currently be under resourced and there would be problems.
You are left with 2 choices, expand St Pancras and staff it to cope with demand or open a second terminal. As I have said in another thread I think not linking HS1 and HS2 was a missed chance, as the second terminal could have been in the Midlands, and the infrastructure would mean high speed running all the way, but it wont happen now. So I think the only option would be to somehow expand St Pancras, which will be easier said than done. Is there any possibility of rerouting SE services anywhere else? Then use that space. Or maybe go upwards, creating space above the platforms. Not easy to see an easy solution.
Regarding Waterloo as the second terminus surely the issue is the long drag on low speed 3rd rail electified lines across south London, longer journey time = less attractive unless you need Waterloo for onward travel, and poorer unit and staff utilisation because of it.