What you're saying make sense, but it doesn't help me understand the connection between people dying in WWI and the lack of recent textbooks. The two people the article describes as having the initial conversation were 30 and 28 years old (give or take a year each) in 1934, which is fairly young for a professor by modern standards, but this means they also presumably had been doing mathematics for around a decade each at this point. Did they not have any older colleagues at all who could have written textbooks? Why didn't they write textbooks in that time period when there would have been ones published in other languages outside of France? I'm not doubting the accuracy of the history being given here or in the Wikipedia article; like I said in my initial comment, I just don't think it's being explained very well, because there's still a gap for me between "lots of people died in the war" and "15 years later there are no textbooks".
I just put in your query (i.e. "Bourbaki was founded ..., young university instructors were forced to use dated texts") into Google AI Mode to explain with references/links/citations which it did reasonably well. The main points are;
1) It was in late 19th century/early 20th century that the german (and other european) mathematicians (Hilbert et al.) revolutionized rigour in the teaching of mathematics with the establishment of definition/theorem/proof (DTP) "new math" style which we understand today as the norm.
2) WWI (1914-1918) wiped out over half the french mathematicians from the various top french mathematical training academies. This was the "lost generation" creating a demographic gap of mathematicians who were trained in the new style above.
3) Thus in mid-1930s you had only a small group of young french mathematicians (Cartan, Weil etc.) who were exposed to the rigorous style now pushed into professorship positions. However their previous generation being lost in the war meant most books were from 19th century written using the "old style math" (i.e. less rigorous). Eg. Standard french university textbooks on Calculus/Analysis were by Edouard Goursat rooted in old 19th century concepts. Hence they are frustrated and start writing books under the Bourbaki name to reestablish french mathematics rigorously.
So a generational crisis (i.e. loss in war) happens when mathematics undergoes a fundamental shift. This leads to books becoming outdated. Young french mathematicians inspired by german abstract school set about unifying all of mathematics using the new rigorous style in terms of various abstract mathematical structures.