They decided to fire Southgate after failing to bring England to the championship. That was not the only reason for them, but they decided this was based on the poor performance shown by England under Southgate, so they decided to sack him.
You have not gotten your facts correctly because Garreth Southgate was not sacked; rather, he resigned from the job to give another person the opportunity to see if they can help England achieve the target they have for themselves after coming close twice to winning it. It was a more honorable thing to do than to allow his record have that he was sacked as England's coach. He also knew that the England supporters were already becoming fraustrated with him. In some games in the Euro where England performed less than expected, the supporters threw empty cups in his direction.
You are right, Gareth Southgate resigned as England's manager instead of being sacked.
He took England to the EURO final three years ago and again to the EURO final in 2024 but the fact is, the Three Lions never won.
And I think, resignation is the best choice for everyone, it's time for someone else.
Several names have been mentioned as potential replacements for Southgate, including Graham Potter, Eddie Howe, Pep Guardiola, and Lee Carsley.