The statement by the WHO reflects uncertainty, but also a comparatively advanced knowledge of what is actually happening. We know that there is a virus that causes the disease and we know that people develop antibodies in response to that virus. We know that immunity is a very likely possibility, we just aren't sure if it's guaranteed.
The situation in 1918 was very different. The overall uncertainties were much greater. The discovery of antibodies was much more recent and our understanding of them more rudimentary. Influenza viruses were not discovered until the 1930s. The efforts to develop an influenza vaccine in 1918 still assumed the disease was caused by bacteria. (EDIT: However, as @JonathanReez shows in the comments, the general concept of immunity through exposure was a possibility known since ancient times.)
If one looks at research conducted shortly after the fact, scientists were only beginning to suspect that people who got the 1918 flu in one wave probably became immune to it in the subsequent waves. Statements about immunity are notably tenuous. For example:
Concerning the important question of the immunity conferred by an attack of influenza, the evidence is not conclusive [...] Parsons, from his study of the last epidemic, inclines to the view that an attack of influenza in the earlier years of the epidemic conferred a considerable but not absolute immunity in the later outbreaks.
(EDIT: Military leaders had similar suspicions based on their observations and experience. Here is an article that documents this in the US context.)
It would be interesting to look at more general public discussions from 1918 about if and when the high mortality was going to end. Based on past epidemics, people may have assumed it would end soon. But I don't expect you will find statements quite like the one quoted in the question because it was not widely assumed in the first place that antibodies would protect from re-infection.