From what I gathered, it is absolutely a question of licensing.
Wordpress is gpl so anybody can host it and provide hosting for others. WPEngine does make money like that and that seemed to rub matt the wrong way.
So he requested more involvement from wpe, claiming they do none (factually false). Then they started trying to extort wpe using trademark as an excuse (even changing the trademark page to reflect their new stance), while also smearing them on the official channels of the project. I'm also skipping the childish behavior of blocking anybody that could be related to wpe from interacting on the official wp.org site, then using this self inflicted wound to say one of WPE plugin is insecure (because they couldn't publish the update) and doing an hostile takeover of it.

WPE is totally within their rights to use the wp software without giving a cent back to Matt. If WP wanted payment for the infrastructure they provide on the .org site, they can change the rules to require commercial entities to pay for it (which they totally could do, but that would hurt the other players in the ecosystem). If they had a (real) trademark issue it would have been resolved under closed doors by lawyers.

there’s an expectation of contributing back or, at the very least, not exploiting the resources of a non-profit.

No, there are no expectations of contributing. It would be worded in the license otherwise (the only expectation of giving back in the gpl is that you publish the changes you did to your users). And WordPress.org is not the non profit but is run by the for-profit company that Matt is the CEO of.

So let’s focus on the actual problem: a large company exploiting a shared ecosystem to run a commercial service.

Like automatic is doing with wordpress? Don't they profit from other devs/companies publishing plugins for them to use on their platform? (Actually not opposed to that, that's the game of open source, but it's a bit hypocritical to only cry when it doesn't serve them).

Edit: From your original comment:

They could have easily avoided this situation without turning it into a turf war.

Yes, then why did Matt turn this into a turf war? He totally expected the community to take his side and turned it into a shit slinging show for all of us to enjoy.