When taharah was a priority stone kelim were used. So for example in the preparation of the ashes of the red heifer,
Mishna Poroh 3(2)
ומביאין שוורים, ועל גביהן דלתות, ותינוקות יושבין על גביהן, וכוסות של
And oxen were brought, and on their backs [were laid] doors on top of
which sat children with cups of stone in their hands.
But for other uses, earthenware (=pottery) could be used as you suggest.
For example the meat of the sin offering could be cooked in earthenware pots Vayikro 6(21)
“An earthenware vessel in which it is cooked shall be broken”
In another example Bamidbor 5(17) the Torah requires an earthenware vessel for administering the waters for the suspected adulteress.
“The kohen shall take holy water in an earthen vessel”
An earthenware vessel that becomes tomeh has to be broken as it says in Vayikro 11(33)
But any earthenware vessel, into whose interior any of them falls,
whatever is inside it shall become unclean, and you shall break [the
The Leviim would not want to become tomeh because they could not enter the Temple in that state. Some people (even Yisroelim) were careful to eat all their food in a state of taharoh (see this Wikipedia article and so they might want to limit their tomeh vessels to those essential for their (inevitable) days of tumah.