Which means he hired the wrong manager.
No.
It means that they were no longer deemed to be the right managers at the time that they were sacked.
You seem to have the notion in your head that a managerial appointment must be permanent; that if a manager is sacked at any point, it is a commentary on his entire tenure rather than the consequence of more specific, more current events; that a failure at the end must also imply a failure at the beginning and throughout the middle.
Take Nigel Adkins at Southampton, for instance. He took Southampton from League One to the Premier League in two years. And then he was sacked. Was he the wrong manager? By your criteria, yes. And yet any sane person would agree that he was absolutely the right manager at the time; that he did a great job. But Nicola Cortese, rightly or wrongly, judged that Adkins had reached his level; that Premier League survival and consolidation would be more likely under a new manager. And so Adkins was sacked and Pochettino appointed.