Yet Southgate, who took over from Sam Allardyce after one game in Group F and has sealed top spot with a game to spare, said the only thing that mattered for now was that England had completed their first task of making it to Russian Federation.
'We were out there talking a few times, trying to keep each other going, to keep the pressure on in the second half.
'They got exhausted, they came out a bit trying to get the win and we took advantage of that.
And Alderweireld, who feels no other striker is better than Kane in the Premier League, believes the England global has set his sights on becoming the best in the world.
This did improve after the break but all in all, this wasn't the kind of display that had anyone believing England stood a realistic chance of getting near to winning the tournament in Russian Federation next summer.
" They are suffering the consequences of 25, 30 or 40 years (of failure in major tournaments) and that's not their fault."
Kane, who scored 13 goals for club and country last month, said England had plenty of leaders in the side irrespective of who wore the armband.
To make the World Cup qualifying play-offs, Ireland must beat Moldova tomorrow night and win in Cardiff against Wales on Monday.
"It's not a relief, I always thought we'd achieve the objective (of qualifying)". 'We will need to defend on him really strictly, maybe sometimes put two defenders on him, ' he said. Now we have to evolve the team.
Southgate admitted that the real work starts now, and that is why he will not overly change the team for what is now an inconsequential game away to Lithuania on Sunday, because he wants to try and develop that fluency. "But we do have goalscorers".
"They've done OK. We haven't learned anything new", he told Omnisport.