Maybe you're using the older Java based libreoffice on the other machines? It was mostly rewritten in C++.
And it seem to be the start up process that differs, as putting them all in a ram-disk does not alleviate the issue, and restarting the app cuts the time in ~1/2 but equally for each distro.
My guess, as I said first is what default libraries as loaded, and possibly how they are configured. I do however find it strange that this has not been mentioned elsewhere as I've been struck bu this difference for years, when I happen to load a pure Debian install (not what I usually use).