1. They don't know the Sith are back at that point
Qui-Gon has had only a brief encounter with Darth Maul. He thinks that might have been a Sith but there's very little proof. In fact, Qui-Gon and Obi-Wan were sent back with the hope than since Amidala was obviously the target, another encounter might help them get more info about the attacker.
QUI-GON : ...my only conclusion can be that it was a Sith Lord.
MACE WINDU : A Sith Lord?!?
KI-ADI : Impossible! The Sith have been extinct for a millenium.
YODA : The very Republic is threatened, if involved the Sith are.
MACE WINDU : I do not believe they could have returned without us knowing.
YODA : Hard to see, the dark side is. Discover who this assassin is, we must.
KI-ADI : I sense he will reveal himself again.
MACE WINDU : This attack was with purpose, that is clear, and I agree the Queen is the target.
YODA : With this Naboo queen you must stay, Qui-Gon. Protect her.
MACE WINDU : We will use all our resources here to unravel this mystery and discover the identity of your attacker... May the Force be with you.
The matter only became settled when Darth Maul had been killed by Obi-Wan.
MACE WINDU : There is no doubt. The mysterious warrior was a Sith.
2. The debate wasn't over the need to train the Chosen One, it was over whether Anakin was the Chosen One.
The Council agreed with Qui-Gon that Anakin was very strong in the Force, but they didn't belief that he was definitively the Chosen One.
YODA : ...Correct you were, Qui-Gon.
MACE WINDU : His cells contain a high concentration of midi-chlorians.
KI-ADI : The Force is strong with him.
QUI-GON : He's to be trained, then.
MACE WINDU : No. He will not be trained.
QUI-GON : No??!!
MACE WINDU : He is too old. There is already too much anger in him.
QUI-GON : He is the chosen one...you must see it.
YODA : Clouded, this boy's future is. Masked by his youth.
After Qui-Gon is dead:
YODA : Confer on you, the level of Jedi Knight the Coucil does. But agree
on you taking this boy as your Padawan learner, I do not.
OBI-WAN : Qui-Gon believed in him. I believe in Qui-Gon.
YODA : The Chosen One the boy may be; nevertheless, grave danger I fear in
Yoda still doesn't believe that Anakin is definitely the Chosen One - the Council only agrees to train him because Obi-Wan states he'll follow Qui-Gon's last wish even if the Council forbids it. Possibly the Council thought it would be better to be able to oversee Anakin's training, since Chosen One or not, he was very powerful, than risk Obi-Wan taking off with him to the unknown.
3. Readings of the Prophecy
While the best known version of the prophecy, as known to the Jedi, went:
In the time of greatest despair, a child shall be born
who will destroy the Sith and bring balance to the Force.
there was scholarship on the fact that it was fairly probable that the prophecy dated to the First Great Schism, thousands of years before the formalized existence of the Sith order; and the reference to the destruction of the Sith was added only later, by Jedi Masters to 'clarify' what they thought it meant. So it is also probable that on the Council, at the very least Yoda was aware of this theory / item and hence did not see the Chosen One -> Sith connection as a definite given.
Additionally, without the 20/20 hindsight, thanks to extra-Dark Side clouded Force, the period around the time of Anakin's birth would hardly seem like a time of 'great despair' to the Council at the time of TPM. That era could possibly be seen as the time of greatest despair due to the behind-the-scenes plots that Darth Sidious and Darth Plagueis were busy setting up, which eventually led to the wiping out of the Jedi, the fall of the Galactic Republic and the rise of a Galactic Empire ruled with fear by the Sith, and depending on EU, the actual birth of Anakin himself.