>why didn't Cover jjust let Vesper make a statement saying they are on a "break" like with the other suspensions they "definitely" gave to the other Holos behind the scenes, so as to save him from this speculation?
Because they have lost trust on Vesper following orders now. Think about it, Cover has total control over all the accounts if they really want to, they could put pull access from Vesper and puppet his account to say whatever they want to preserve the image that things are fine. If they wanted to make it seem he was sick, they can post on his twitter daily updates. They can also have him keep on liking and retweeting stuff. Making an announcement using their main account to suspend him instead of using Vesper's prevents a chance of Vesper contradicting their statements when he comes back or even outright saying Cover took control of his accounts to lie.
You could also say Cover learned from the Rushia incident to not give their talents the benefit of doubt if they have done something seriously out of line. Rushia was able to send chatlogs to dramatubers when she should have been on break as well as posting on her roommate account that kept making it look like Cover was controlling her life despite the statement they put out on respecting their private life and relationships. Cover had to make a statement fast and quick before Vesper could plan out something that would muddy the waters either from his Homo or roommate account to paint things on his side.