> If that were true, the FSF wouldn't call it a free license.
It is true; the license gives you the source, to do with as you please, including closing it off.
Famously, Microsoft included BSD licensed tools in Windows since the 90s and did not distribute the sources!
And that is completely legal. If you want to force the users to distribute their changes to your open source product when they are redistributing the product, you need to use GPL.
MIT/BSD licenses are pro-business - any business can take the product, change a few lines and redistribute the result without making their changes available.
GPL is pro-user - anyone who gets the source, makes changes, and then redistributes the result has to make their changed sources available as well.
The FSF has written extensively on why (in their opinion) you should prefer copyleft licenses over non-copyleft licenses, but they don't require a license to be copyleft in order to be considered free. It's worth spending a bit of time on their site to understand their point of view. Just be careful not to drink too much of the Kool-Aid or you'll become one of those annoying people who never shut up about the GPL on forums.
> you should prefer copyleft licenses over non-copyleft licenses,
For most, but not all, software. Stallman did famously argue for libvorbis, which you may know as the ogg codec used mostly by games and spotify, to be licensed under BSD instead of the (L)GPL.
True, there are exceptions. Stallman thought strategically. Having a free-but-non-copyleft licensed reference implementation is necessary if you are trying to wrest dominance from an established but proprietary standard.
But I'm willing to bet that he'd have pushed for GPL if he wasn't trying to topple MP3.
You are thinking of copyleft (e.g. GPL)