I found this interesting passage from this Wikipedia page:
There are several rules in the Theravada monastic code by which a
bhikkhu is "defeated" - he is no longer a bhikkhu even if he continues
to wear robes and is treated as one. Every ordination ceremony in
Theravada Buddhism is performed by ten bhikkhus to guard against the
possibility of the ordination being rendered invalid by having a
"defeated bhikkhu" as preceptor.
I've personally never heard of this, but apparently ordination of new monks are done by multiple monks, to ensure that at least one of the monks performing the ordination is not "defeated". This is to ensure that every ordination can be traced back all the way to the Buddha. That said, I doubt that a detailed record of the lineage is kept.
I would say that for Theravada Buddhism, the lineage of monastic ordination is important (without keeping meticulous records of lineage), but not the lineage of teaching.
As the Buddha has stated in the Mahaparinibbana Sutta (quoted below), he is not leaving behind any successor. Instead, his followers must depend on themselves, with the Dhamma (the Buddha's teachings) as their basis. There is no mandatory need for a teacher as an intermediary (although the Buddha did specify criteria for a Dhamma teacher in the Udayi Sutta). The Buddha also did not hide any esoteric teachings.
Due to this, a lineage of teaching is not important.
Thus spoke the Venerable Ananda, but the Blessed One answered him,
saying: "What more does the community of bhikkhus expect from me,
Ananda? I have set forth the Dhamma without making any distinction of
esoteric and exoteric doctrine; there is nothing, Ananda, with regard
to the teachings that the Tathagata holds to the last with the closed
fist of a teacher who keeps some things back. Whosoever may think that
it is he who should lead the community of bhikkhus, or that the
community depends upon him, it is such a one that would have to give
last instructions respecting them. But, Ananda, the Tathagata has no
such idea as that it is he who should lead the community of bhikkhus,
or that the community depends upon him. So what instructions should he
have to give respecting the community of bhikkhus?
"Now I am frail, Ananda, old, aged, far gone in years. This is my
eightieth year, and my life is spent. Even as an old cart, Ananda, is
held together with much difficulty, so the body of the Tathagata is
kept going only with supports. It is, Ananda, only when the Tathagata,
disregarding external objects, with the cessation of certain feelings,
attains to and abides in the signless concentration of mind, that his
body is more comfortable.
"Therefore, Ananda, be islands unto yourselves, refuges unto
yourselves, seeking no external refuge; with the Dhamma as your
island, the Dhamma as your refuge, seeking no other refuge.
"And how, Ananda, is a bhikkhu an island unto himself, a refuge unto
himself, seeking no external refuge; with the Dhamma as his island,
the Dhamma as his refuge, seeking no other refuge?
"When he dwells contemplating the body in the body, earnestly, clearly
comprehending, and mindfully, after having overcome desire and sorrow
in regard to the world; when he dwells contemplating feelings in
feelings, the mind in the mind, and mental objects in mental objects,
earnestly, clearly comprehending, and mindfully, after having overcome
desire and sorrow in regard to the world, then, truly, he is an island
unto himself, a refuge unto himself, seeking no external refuge;
having the Dhamma as his island, the Dhamma as his refuge, seeking no
"Those bhikkhus of mine, Ananda, who now or after I am gone, abide as
an island unto themselves, as a refuge unto themselves, seeking no
other refuge; having the Dhamma as their island and refuge, seeking no
other refuge: it is they who will become the highest, if they have
the desire to learn."