As per scriptures, it is necessary to wear an upper-garment while participating in sacred rituals.
He shall carry a bamboo staff and a pot filled with water; wear a double
sacrificial cord ( this verse is repeated in Apashthamba too)
, a turban, a skin as an upper garment, and shoes; carry an umbrella; maintain the sacred domestic fire; and offer the
new-moon and full-moon sacrifices.
Note that, wearing the sacred thread (for those who are initiated) and an upper garment (for everyone) are both necessary.
He should wear an upper garment during these five activities: private
vedic recitation, voiding urine and excrement, giving a gift, eating,
and sipping water.
And, the upper garment has to be worn in a particular fashion as stated in the following verse:
When he is paying his respects to elders , offering sacrifices and
softly reciting prayers;* and when he is eating, sipping water, and
doing his private vedic recitation, he should wear his upper garment
over his left shoulder and under his right arm
Ideally, the upper garments are made from skins of certain animals as stated below:
A Brahmin’s girdle should be made of Muñja grass, Ksatriya’s of a
bowstring, and a Vaisya’s of hemp thread. Brahmin’s upper garment
should be the skin of a black antelope, a Ksatriya’s the skin of the
spotted Ruru antelope,
However, as a substitute for the upper garment made from skin, using simply a cord (or any piece of cloth like a towel for example) is allowed as per the following verse:
He should always wear an upper garment or, in place of that garment,
wear just a cord slung over his left shoulder and under his right arm.
So, it is not that the shawl or the towel or any piece of cloth becomes the substitute for the sacred thread. It is simply playing the role of an upper-garment, wearing which in a particular manner is mandatory in religious rites.
Now, the persons who are not wearing sacred threads are also allowed to participate in religious rites. But the rule of wearing the upper garment in that particular fashion still remains the same for them.
AD- The Dharmasutras of Apasthambha
BD- The Dharmasutras of BaudhAyana
VD- The Dharmasutras of Vashishtha.