None of that gives the right to chop down someone else's tree on some else's land. The reasonable course of action was to contact the tree's owner and to cordon off the area "at risk" in the meantime.
The only possible redeeming aspect is if the tree is part of the "demised land" of the restaurant, i.e. land that is part of their lease if they are leasing their premises (this is not mentioned in media reports as far as I know so it is unclear), but the reasonable course of action would still have been to contact the owner/landlord first as they usually must give permission.
Trees are already protected because, again, no-one has the right to chop down a tree that does not belong to them. This is why the people who chop down the Sycamore Gap Tree were charged with criminal damage. A tree preservation order adds another layer of protection in that even it is your tree you are no longer allowed to do any work on it without the Council's permission. In this case it is possible that they simply did not think it was necessary as the tree was in a Council-owned park.
The council own the land, and leased it to the restaurant. They claim the Toby Carvery "has broken the terms of the lease which requires Toby Carvery to maintain and protect the existing landscape"
https://www.telegraph.co.uk/news/2025/04/16/toby-carvery-cou...
There's no need to see malice where indifference and incompetence will do. You need to do a box-ticking exercise, you buy in an expert. The expert says you need to do X, you don't press too hard against that. They say they can do it for you. You assume they know what they're doing and say "OK, do it".
We'll have to wait for the courts to find out exactly who said what to who, and who made what decision, but this is about as much as we can infer for now. The tree's still gone.