I have no doubt that Poch told him the truth. But I also have little doubt that Poch gives Harry an enormous amount of leeway to decide when he's fit and when he isn't - seems part of Poch's loyalty to him, and his approach in terms of encouraging belief that his players can beat the odds.
I agree, though - going forward Poch would benefit from being more stern in this area. A bit of short-term pain could mean a longer-term gain.
As for Southgate, I have no idea if he told Kane anything. The FA have a habit of calling up our players to play these utterly meaningless glorified friendlies on a regular basis, regardless of their condition. Winksy is the only exception to that - ironic, because the team could have done with him against the Dutch and the Swiss.