R’ Yochanan explained that the reason for the Mishna’s Shevua is
Because Mamon that is in Safek requires a Shevua to divide it.
Beis Din may not divide things without Shevuos.
To offset the other person’s Muchzakus.
So people won’t go around grabbing other people’s Taleisim, claiming that it belongs to them.
So the other person, who sees that this person is so certain of his position that he is even swearing, will back down if he is lying.
How is our Mishna, that Tallis is divided with a Shevua, compatible with R. Yosi by Muneh Shlishi, who says that all three Manos should be kept until Eliyahu comes because “What will the Ramai lose in his treachery”?
It isn't.
In that case there is a definite Ramai. In our Mishna, maybe both people picked it up at the same time!
In that case, keeping all three Manos will cause the Ramai to lose what is really his. By Tallis — that is, Metziya and Mekach Umemkar — he doesn’t care if its kept, for he is not losing anything of his own.
Beis Din has enough Talesim in Geniza. They don’t want to impound another one.
Manos are worth more than a Tallis, so the Ramai will be penalized harder.
Rabba explained that a Modeh Bemiktzas must swear because
He really wants to lie about the whole amount, but is unable to lie in the face of the person who lent him money.
It’s a Kal Vachomer from Talis, in our Mishna.
So Hodaas Piv will not be greater than Aidim.
Of a Tzad Hashuveh of Korban and Aid Echad.
So people won’t go around grabbing half of other people’s Taleisim, claiming that this half belongs to them.
“Ishtamutei…” (He wants to slip away…) means
Someone who owes money wants to hide himself so he won't meet his debtor.
Someone who owes money denies the debt outright, because he doesn’t have the money now.
Someone who owes Reuven money will try to “slip away” from paying other obligations, so that he’ll have money to pay Reuven.
Someone who owes money tries to “slip away” from the witnesses who can testify about the amount he owes.
Someone who owes money says that he owes less, because he doesn’t have the money now.