If there really were "a war to end all wars", would civilization stagnate?

Yes, because the only war that could end all wars would also have to end civilization (termination being the most extreme form of stagnation).

I don’t agree with your reasoning, though, because you seem to be confusing “war does X, Y, and Z” with “war is the only thing that does X, Y, and Z.” I also disagree that wars cull out the violent people. These days, the most violent people seem to be the ones calling the shots, not the ones taking them.

What drives war is exploitation. The idea that some people ought to be lords, and others should be eager to serve them and struggle to survive, or be overpowered in whatever way that takes. And that the lords should strive to amass as much power and wealth as they can, endlessly.

That idea exists even in “peace time” capitalism (and corrupt Communism in e.g. the USSR and China) – groups of corporations and mega-wealthy investors play a game to consolidate wealth, favorable laws, and other forms of power (e.g. obedient politicians, CEOs and journalists), not only to the extreme exclusion of others, but also with little/no regard to the destruction of the planet’s ability to support life as we know it.

Changing that pattern of thinking and behavior is the only thing that could end all wars.

And that pattern of thinking is in the assumptions of this question’s wording. Our Economic “Science” defines ideas such as “productivity” and “stagnation” as related to the limited measures of how much the lords have increased their wealth.

If there were to be a war to end all wars, something would take place during that war to effectively render to all humanity something akin to frontal lobotomies.

There is a good chance it would.
There are only 3 main drivers of technological innovation:
1. War.
2. Glory to gods and kings.
3. Money.

You barely have anymore kings, and capitalism prefers minor iterations over revolutionary advances, because of the risks involved.
It reminds me of that hypothetical board meeting about funding for a manned Mars mission, told by Neil deGrasse Tyson.
Q: How much is it going to cost?
A: A lot.
Q: What is the return of investment?
A: I do not know, probably nothing.
Q: How dangerous is it?
A. Very.
That meeting is over after 5 minutes.

Who cares if it did? If we are content why isn’t it enough? I think if we all felt safe and cared for we would still get technological advances and there would still be some competition to achieve or win it be first, so that would still be a driver.

I appreciate the simple things in life so much more having been in chronic pain for many years and finally getting significant relief. However, if I could trade being more mature and generally more grateful and appreciative, for never having suffered that shit, and the ongoing medical crap I deal with, I would trade in a second, and maybe you could say willingly be more stagnant.

War will never stop.
No matter how civilized we want to think we are, there are primitive aspects ingrained by nature we will never shed.
Some of us will believe we are better than all others to the point it breeds angst.
We will always have people embroiled in fury, who cannot be stopped, short of death.