You've exposed the flaw in your argument though. International football is nothing like club football. What other trophies has Southgate got the option to win if he fails to win the prize we all want? It's the Euro's and the World Cup. Nothing else matters. They don't come along often and when they do there are other good sides going for them too. Southgate is far from the best manager out there, but there have been plenty of very good managers who have failed at International level and when it comes to England it doesn't seem to matter what pedigree you have, you'll fail. Scaloni has taken Argentina to the Copa America and now World Cup final with no pedigree at all. It's his first job! Meanwhile Luis Enrique and Hansi Flick who have excellent pedigree have failed to impress with their sides. Mancini won the Euro's then couldn't qualify for the World Cup.
There are much better managers than Southgate, but you don't know if they'll do any better because it's a completely different game to club management. You have to be confident in your system, confident in getting shat all over by the media for every decision you do or do not make, and able to create a team and squad that play for the shirt and each other and works successfully.
I want Southgate to go now, but I don't know who replaces him. I'd personally go foreign just because I think Potter and Howe are still too raw and unobtainable at the moment. I think times have moved on and English football is more aligned to things on the continent and foreign managers would easily understand our culture. Pochettino is available. But he doesn't have much pedigree when it comes to winning things. We can debate Southgate all day long, but what's the solution? Who comes in and does a better job?