Dear Wrongan ji,
I think it is correct as you said, but in custom I think this has been changed and stopped with karthas and gnathis and sahagothras. Anyhow, what I personally feel is only the karthas and gnathis have the rights to eat that also since they are the one involved in the karma and also in one or other way related to the respective pithru. How the sesham should be shared then with near and dear who does not have any sort of even knowing about the respective pithru?? I think this should be clarified even better to know it fully.
Pranams
What Shri Durgadasan states seems to be the present prevailing practice (sagOtra not sahagotrA). But the ancient smritis allowed a different practice, as may be seen from the following extract of The History of Dharmasastra by Mahamahopadhyaya Dr. P.V. Kane in the section dealing with "SrAddha":
He should then make according to his means presents (to the brah-
manas) of cows, land, gold, clothes, splendid beds and
whatever else was liked by the brahmanas or by the perfor-
mer Inmself or his father. He should not be stingy in his
presents. Then he should request the brahmanas to say
'svadhA' and they should do so. Then he should request the
brahmanas to pronounce the following benedictions and should
receive them from them while he faces the east. They are 'May
the pitrs be kind (not dreadful) to us' ; the brahmanas should
say 'so be it; 'May our family increase'; they should
say 'so be it'; 'May donors in my family prosper and also
(the study of) the Vedas and progeny and may these benedictions
come out true'; they should respond 'so be it'. He should
then remove the pindas and request the brahmanas to utter the
word 'svasti' and they should do so. The remains of the food
eaten by the brahmanas remain (unremoved & unwiped) till
the brahmanas leave. The remnants of food that lie on the ground
are the share of the group of slaves that were straight-forward
and not roguish (dishonest or shirkers). The performer holding
a vessel full of water and muttering the
verse 'vaje vaje' (Rg. VII. 38 8, Vaj S. IX 18, Tai S. I.7.8.2)
should touch the brahmanas with the tip of kusas and allow them
to depart. He should follow them out of his house for eight paces,
should circumambulate them, should then return with his
relatives, sons and wife, should then perform the daily Vaisvadeva
and balihoma. Then after Vaisvadeva
he should together with
his relatives, sons, guests and servants partake of the food
that remains in the cooking pots after what was eaten by the
brahmanas.
The words "ishTai: saha" reflects this position, I feel.