Temple Mills Facility Description available here:
https://highspeed1.co.uk/media/kpfhrm2h/eil-tmi-sfd-final-2022-v1.pdf
Which has the following info: Number of tracks which are completely electrified
x22 (inc. wheel lathe road) of 400m (my bold). Given the limitations on other maintenance roads (e.g. only 6x roof inspection roads, only 2x CET equipped roads) it is reasonable to assume that at least some roads need to be left clear for shunting at any time.
It is also relevant that Temple Mills explicitly cannot undertake 'heavy' maintenance and so any operator would still need a second depot to undertake the jobs beyond the capability of Temple Mills.
From satellite imagery, the space exists for the 2x short roads to be extended to 400m length but not within the current boundary fence of the site (which could of course be moved, subject to wrangling between HS1 Ltd and NR). However those roads would require additional shunting to access what appear to be the cleaning roads. Remember Depot capacity isn't just about how many vehicles you can physically fit, but the ability of the depot to do all required tasks.
Stabling at St Pancras should be possible, from RTT Eurostar already do.
At Stratford the layout raises it's ugly head again - stabling in the 400m platforms would require a lengthy wrong-road move at one end of the day or the other if the train is leaving/entering service at St Pancras, or to run down to Ebbsfleet and back. The question of where the driver depot would be is also present here.
At Ebbsfleet the layout isn't as problematic but the assumed lack of a driver depot would be an issue (although not unsolvable).
Dagenham is unlikely to be used as a stabling point, there's barely enough space for the sidings as it is and they're used for accessing the freight terminals there. It's also not accessible to HS1 trains from the St Pancras Direction, so trains would need to be taken out of service at Ebbsfleet.
Stabling at Ebbsfleet or Stratford Int'ls would also then require track maintenance to be accommodated. Certainly not impossible, but an additional issue to overcome. The HS1 Sectional Appendix does explicitly allow stabling of trains in St Pancras, Stratford and Ebbsfleet Platform Lines as per the relevant UK Rulebook modules. Also each station only has 2 platforms to stable trains at, which is quite restrictive - especially if either operator wanted to call at the stations early in the morning or late at night.
Tbf Eurostar aren't claiming there's a lack of space in the UK, just in their depot. I do agree that they're going to find it hard to defend their claim that no solution to the depot question is possible though. However the question of depot capacity is also dependent on the service plan for either prospective operator. If they need trains ready for the first morning services from London that's a different problem to if they're planning to do the maintenance in the afternoons between a morning and evening run.
Within all this is the business side of things - a new operator is not going to want to drive costs up by scattering trains up and down the length of HS1, because then they'd be unable to compete on price with Eurostar.