He has good bits (very good team atmosphere, man manager, has worked with the younger generation in the under 21s), and bad bits (average tactician, very bad in game management / subs, questionable first XIs, very poor record against the traditional big European sides).
I think though above all by far the biggest quantifiable difference from now and the previous generations, is just the wealth of attacking talent available. For example in that generation you listed, where we had great defense and midfield, but our attack was not as good. In 2006, we had Lennon, Cole and an injured Rooney. In the 2006 - 2016 era, we had some shocking lineups, with the likes of Welbeck, Walcott, Sturridge even Heskey getting regular starts.
Now we have so many attacking options that can't even get into the squad, that are equal or better than the ones above that started previously (e.g. Toney). The question is then, do you take the gamble on someone who'll use those attacking options? Or persist with Southgate's good atmosphere, but potentially waste the best attacking group of players (as a whole) in 30+ years? The tactical options he made both against Italy and France, were very poor. Personally for me I don't see him suddenly knowing how to attack teams or making use of the attacking ability we have, when his default setting is to start Sterling and Mount whenever they're fit. He's not going to get sacked until we fail again at the Euros against any good side we come up against.