Why has "using" gone out of fashion?

In the US, anyway, using those words instead of "using" has been basically standard for a whole lot longer than 10 years in the business world. My memory is that it start in earnest in the '90s, although it's possible it was earlier.

It's just people trying to sound smart.

This I think is one of the many wrongs perpetrated by “business speak”. People say “invite” for “invitation”, then instead of using “invite” as a verb they say they “sent an invite”. They don’t say “installation” and say rather “install” as a noun. They say silly trite things like “circle back”. They load up conversations with sports metaphors and military jargon. A layoff or staff reduction is now a “reduction in force”. A couple of decades ago it might have been known instead as “resizing”. “Taking point” on a project has become as common or more common than “leading” it. I think you’ll also find more “utilization” in use than “usage” - which in some contexts has a slightly different connotation but usually the shorter word will work just fine.

It’s just transient fashion, driven by people who don’t read books. In another few years the same type of people might be using “employ” or something instead of “utilize”, or returning to “use”.

If you want your English to be good, try to spend more time with books, and less time with anything written after about 1960. This (excellent, and free) advice applies to native speakers as well as those enjoying English as a second (or third, etc.) language.

Also, interestingly enough, "use" and "utilize" are not exactly synonyms, even though they're usually used as if they were.

Technically, "use" means to use something for its given purpose, and "utilize" is a specific kind of using. It means "making the most effective use of something, often in a manner not initially intended."