In 21, 3 the Rama wrote:
שלא ליתנו לכובסת גויה .
Not to give Talit with tsitsit to a Gentile washwerwoman {Mishna Berura: instead, remove the Tsitsit before}.
In Gemara, there seems to be an additional problem (Menachot 43a) {note that the cases of Gemara are when a man wants to sell a Talit, which the Shulchan Aruch treats in 20, 2, not when he wants to give it to a gentile for washing as in the case above in Shulchan Aruch 21, 3}:
And [this is so] not withstanding that they said, A man may not sell a garment furnished with zizith to a gentile unless he removed the zizith. What is the reason for this? - Here it was explained, on account of a harlot {Rashi:this Gentile will enter in relationship with a non-religious Jewish woman and they will marry}. Rab Judah said, It is to be feared that [an Israelite] might join him on the road and he might kill him {Rashi: he may use the Tsitsit to gain confidence of a Jew and kill him}.
Now consider SA OC 15, 1:
מותר להתיר ציצית מטלית זה וליתנם בטלית אחר אבל שלא ליתנם בבגד אחר לא.
It is permissible to remove tsitsit from this talit and fasten them to another talit; but to not fasten them to another talit, no.
The second part of the statement is an extract of Tosfot Shabbat 22a (Dibur Hamatchil Rav amar).
The reason is that now a mitsva will be cancelled from the Talit without a new mitsva for another Talit. The source is Gemara (Menachot 41b; Shabbat 22a): In Gemara, Rav said it's prohibited to move Tsitsit from garment to garment, and Shmuel allowed it. The Gemara for Halacha follows Shmuel's opinion . Tosfot Shabbat 22a explained that Shmuel thinks that Tsitsit is "a duty of the garment, not of the man". Therefore, one is not allowed to remove them from a 4 Kanfot garment without other purpose. The problem is "mitsva's cancellation". So Magen Avraham explained that if the garment will be sold to a Gentile, anyway the garment of a Gentile has no duty of Tsitsit, so the vendor is allowed to remove the Tsitsit before. when giving the Talit to a whasherwoman (21, 3), the Talit remains duty, but the purpose is to fast a new tsitsit after laundry, so Mishna Berura allows. Mishna Berura on Siman 15, cited in the OP adds a reason (add a reason after the reason of cancellation of mitsva): disgracing Mitsva. (I guess that an example who need this reason is if a man fastens and removes tsitsit from a garment as a game, without purpose.)
Obviously, for your question, there is no cancellation of the mitsva and no disgracing if your intention is to fasten on this Talith a tsitsit (this tsitsit or new Tsitsit): maybe even according to Rav it is allowed. The SA OC in 21, 3 is perhaps not linked with your question. Magen Avraham 15, subparagraph 2 said in name of the Levush that one is allowed to remove tsitsit when his purpose is to fasten a more beautiful tsitsit.
Thus, to your question:
if I want to put my tallit (either gadol or qatan) in a laundry machine, am I allowed to remove its ẓiẓit beforehand
--> Yes.
