I think the Championship is a good idea but very flawed, even before the pandemic. It's classic ICC.
I think teams not playing every team is a flaw. But, realistically, England, India and Australia aren't going to want to take up weeks preparing and playing against Zimbabwe, Ireland, Afghanistan and even some of the more established test nations. Certainly not when that time could be spent having an ODI series against one another. I think the last time England played Zimbabwe in a test match James Anderson was making his debut. So, they aren't in a hurry.
Also I'm not entirely sure if those three smaller nations would necessarily be able to take part from a financial perspective without the ICC stepping in. Even before the pandemic Ireland had to cancel the only test match they had scheduled because they couldn't afford to go ahead with it.
The points system and inconsistent fixtures for each series is a problem. India and Pakistan are not currently going to play each other. I think if it came to 'forcing' India into doing it then the tournament wouldn't happen. And The Ashes isn't going to reduce the number of matches to fit around the World Test Championship.
So have two or three groups, play 2 match series. The winners of each group (if it were 3 groups) could meet for for a 'festival of test cricket' and play each other a couple of times over say five or six weeks. The winner could be declared to be the team with the most spirit of cricket.