The Wachowskis did not want to make another Matrix movie but the studio approached them and said that they would make the movie either way but they would be given full artistic freedom if they agreed. They basically had the choice of doing this themselves or letting the studio turn it into a soulless franchise detached from their own vision.
The movie is mostly an allegory for how the movie was created and why it shouldn't exist. Aside from mindless indulgence the ending is also giving a middle finger to the production company. The philosophy is dull because everything that needed to be said was already said so the only thing left to do is repeat it more blatantly for the audience in the back.
That doesn't mean it's a great movie and you can argue that a movie should be enjoyable without context, but for me knowing this context allowed me to enjoy it through that lens a lot more than I probably would have had I not come into this knowing this.
So they own it as much as the 1st 2nd or 3rd. They did not have to do that. In fact it would have been an even stronger statement if every review of the 4th started off with "done without the original creators input or approval".
But now the 4th has been done with the original creators input and approval, so it's their fault, their mess.
In hindsight you can now see the wonder and amazement of the first movies were not just from them but everyone that worked so hard for them, not just the actors but everyone behind the scenes, even the clever color timing, completely absent from the 4th as well as the horrible music, everyone just phoned it in for the paycheck.
"Because someone else would do it if we didn't" is not a viable excuse, it would have been a statement otherwise, a path they purposely didn't chose for either ego or profit.
The Wachowskis not being involved wouldn't have affected the movie's commercial success. There are plenty of examples of "franchises" being carried on by different directors even in directions the original director never intended and to still be successful.
What the Wachowskis did was take a story that had nothing left to tell and then make a movie about the experience of having to keep adding to that story, drive home the point of the original story and provide an ending to the story that is impossible to write around without discarding the source material and starting over (not that that stopped e.g. Star Wars Rise of Skywalker from doing pretty much this with The Last Jedi).
The Matrix is intrinsically anti-capitalist, anti-authoritarian and queer. But much of its fanbase likes it entirely for cosmetic reasons (cool slomo fights and explosions and sunglasses). WB would have been entirely happy to cater to that crowd and would likely have gotten away with it. Instead the movie ended up amplifying those themes and alienating anyone who just wanted another film with cool slomo fights and explosions and sunglasses. And IMO they did it in an interesting way nevertheless.
