// ... this is an international emergency that dwarfs all else. //
Maybe so, but that's no reason not to change leader if the leader isn't up to the task. In this particular instance it would also be highly relevant who was lined up to replace him, but there have certainly been worse crises the UK has faced, that didn't stop a leadership change. In particular, you could point to the Norway debacle in 1940, with worse to come around the corner; or, to a lesser extent, to the Suez Crisis. In that case, Eden resigned a month or so after the height of the crisis, and at least primarily due to ill health, but as his successor Macmillan wrote, "nature had provided a real health reason [when a] diplomatic illness [might otherwise have had to be invented]", and in any case, the fallout to the UK's prestige was still significant, and it would be odd to point to that as an excuse for his continuing.
The obvious difference between the two cases mentioned above, as opposed to Johnson, is that those resignations were directly connected to the foreign crises in which they occurred. Still, a leader representing the UK's interests abroad ought to have the trust and support of the country, rather than be kept on for appearances' sake.