Both of your alternatives (became and has become) would also work and they wouldn't really change the meaning of the sentence.
Should has many uses as an auxiliary. This is not one of the most common, but it's not exactly rare:
used for describing a fact or event that someone has a particular feeling or opinion about
It’s hardly surprising that people should be suspicious of politicians’ promises.
How sad that she should have no one to comfort her.
Claudia was shocked that anyone should believe such a scandalous story.
It’s odd you should mention Ben – I was just thinking about him.
It's interesting that that definition is in Macmillan's "American English" section. The Cambridge Dictionary marks this use as "mainly UK":
used after "that" and adjectives or nouns that show an opinion or feeling:
It's odd that she should think I would want to see her again.
It's so unfair that she should have died so young.