The build-up to the team's opening World Cup match against Iran was dominated by uncertainty over what the consequences would be if the rainbow-coloured band was worn.
The Football Association had sought FIFA approval to wear the armband at the World Cup in Qatar as far back as September when the year-long OneLove campaign began, but received no reply.
The governing body was prepared to be fined for wearing it, but when it emerged in the hours before kick-off that FIFA could issue yellow cards to Kane and the other six European captains who had been set to wear the band, the plan was dropped.
Wearing the anti-discrimination armband in Qatar, a country where same-sex relationships are criminalised, would have been an especially significant move and the seven FAs involved have faced criticism for backing down to FIFA on the issue.
However, Southgate said after the win over Iran he understood FIFA had to be careful about setting a precedent, and told talkSPORT's World Cup daily podcast: "I was a bit concerned that approval hadn't been given and I think there are lessons to be learned from that, if I'm honest.
"But we had to focus on the football and other people had to do that talking overnight and (on Sunday and Monday morning).
"For me I didn't want anything distracting the players. There's been a huge amount of talk around us, which is understandable.
"If we can help raise awareness of other issues then we want to do that and we will always try to do that, but of course we're here to try and take our supporters on a great journey again."
Kane expressed disappointment at not being able to wear the armband, and his team-mate Jack Grealish said it was "a bit stupid" that he had been prevented from doing so.
Kane wore a FIFA-approved 'no discrimination' armband instead.
Netherlands captain Virgil van Dijk rejected suggestions that the European countries involved had been spineless in backing down over the issue.
"I play in a position where a yellow card is not useful. I became a football player and I want to play these kind of tournaments," he told Dutch broadcaster NOS after his side's victory over Senegal on Monday.
"There are people who say we don't have a backbone, but that's not how it works."
FIFA has been heavily criticised over the matter. It only announced its plan for an approved armband on Saturday, and even then, it stated the 'no discrimination' part of its campaign would only begin at the quarter-final stage. On Monday that position shifted.
It followed a late U-turn on alcohol sales within stadium perimeters last Friday, and has again raised questions over who ultimately is in charge at this World Cup - FIFA or the Qatari authorities.