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.
The license is only three paragraphs long. You can see it does not contain text supporting your claim.
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.
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.
But I'm willing to bet that he'd have pushed for GPL if he wasn't trying to topple MP3.