As a non-England fan, I think he's certainly earned the opportunity to manage for the World Cup, and barring a disaster there, should keep the job.
I think you've all rightly pointed out that he's not going to change, he's not going to become a proactive manager, and he's not going to set up teams to attack. It's got to be really frustrating when you have so much attacking talent and you know it won't be fully utilized.
It definitely is an effective to win games, especially in international knock out tournaments.
That all said, the thing that would really gnaw at me in the back of my head as an England fan is just how perfectly setup this tournament was for England to win. Huge home field advantage, no major injuries, very easy draw, some major rivals (France for example) getting knocked out early, etc. and Southgate couldn't get it done. I know your talent will age more which will be good, but you're likely not going to get an easier run for a title. If Southgate couldn't do it now, I'd worry he'd never be able to.
Still, you need to keep him unless another, clearly better, option says they'd be willing to do it.