**speculation only**
I reckon it probably went something like this.
CTA and EB put together an entry for the competition, probably in a hurry and not necessarily thinking they'd actually win it.
Then it became a real project. At this point everyone has to get a bit serious because it has to be paid for, and as far as I understand this wasn't just a design proposal but a business proposal. It's not like CTA and EB just have to work to produce designs for Lambeth or a developer, they have to fund it as well and it seems like a chunk of that funding was coming from CTA themselves (through that house sale).
And the reality of the project is that it was actually quite a sizeable and complex development that had to be built in a very short time (and the size/complexity probably became increasingly apparent during the period after winning the competition, while putting the planning application together). Something like that is not a small undertaking even for an architectural practice that is just doing the architectural work (let alone also worrying about the business risks). To do all the design work to get it to construction stage, to sort out planning permission and building regs and all the other stuff that you have to deal with is probably enough work to keep two employees at least busy for a good few months, perhaps more like a year.
So putting myself in Carl Turner's shoes, and I don't know him and he probably has a lot more business experience than me and I am just trying to imagine the scenario - I, at that point, would be breaking into a bit of a cold sweat because not only am I partly funding this thing with my own money/house and taking on the risk that the commercial side of things doesn't pan out but I am committing myself and at least some of my salaried employees to a load of work which we only get paid for if the business side of the project works out.
And I don't know loads about EB but as far as I understand they are a fairly small operation whose experience is doing small public space/landscaping projects which no doubt they are perfectly competent to do but I don't imagine they have the experience or resources to get stuck into all aspects of the technical design of something like Pop Brixton.
So with CTA investing a substantial amount of their own money, plus being the ones who would have to undertake the majority of the design work I can see why a 1/3 of 3-way split of decision-making power might start to look like rather a worrying arrangement. I can see why they might start to think they'd rather pay EB in proportion to their design input instead of continuing with them as equal-share business partners.
Of course I can also see how this could be massively unfair to EB, if its true that they were deliberately exploited and if there wasn't a genuine intention that they would be able to continue as equal partners, assuming that was the nature of the original agreement.
And if EB had called the bluff? I expect CTA would have resigned for the reasons above and the project would not have been able to go anywhere without a similar company coming on board which would probably have similar concerns and similarly would not want to go ahead on the terms which EB wanted.
All just my speculation though.
The Buzz article of course overstates the extent to which the final thing differs from what was originally proposed. Not interested in the detail of the bid entry, but the "green oasis" soundbite. Which I actually thought came from some PR attached to the original bid but the link in the article takes us to an instance of its use as PR for what's actually there now.