MySOS is still necessary for everyone.

But there is also a "MySOS Web" version which does not need smartphone use or App download (I think its only for Fast Track countries). With this, you can apply for the MySOS online without download, then register what you need to get the blue/green approval screen, then either somehow show this screen at immigration (I suppose with a phone or laptop) or show a printed copy of it.

(so if you don't want to have their tracking App or worry about airport wi-fi, its an alternative)

Here's the details (scroll down for the Web version info)
https://www.hco.mhlw.go.jp/fasttrack/en/