Throwing out Tallis and Tzitzis

I’d like to confirm what I’ve heard from the Rav regarding throwing out old Tallis and Tzitzis.
Until I met the Rav, for half my life, I had R. Belsky’s psak of "rip off the tzitis into shaimos and you can throw the tallis/tzitzis in the trash.

Then, I became a follower of our Rav, and even the tzitzis went into the trash!
Is my mesorah correct?
And while i’m at it, mezuzah covers, old tfilin bags, can i trash them?

Old Tallis and Tzitzis (known as Tashmishei Mitzvah) may be thrown out not Derech Bizayon. Practically, you can put them in a bag and place the bag in the garbage.

Old Mezuzah covers and Tefillin bags are a different story. They are Tashmishei Kedushah and may not be thrown out. This is assuming the bag is designated for the Tefillin only. If you use it to store other things as well, that’s a different story. For example the larger bags that many use for Tallis, Tefillin, some loose change, papers… you name it, isn’t considered Tashmishei Kedushah.

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