How can there possibly be a best band of the 60's or any decade for that matter?!
Decades are a fun and mostly valid way of classifying music, best means 'my favourite and I'm prepared to say why'. Easy.
If the button can have the Stooges, I'll say Can (although they did loads of great stuff in the 70s too).
)Sly & the Family Stone
So it's not 'who mwere the best' then but' who was you're favourite' from [insert decade]?! An entirely different question as one seeks some objective definite answer whilst the other is entiorely subjective.
They played on and produced hundreds of records, including classics like Walking the Dog, Hold On (I'm Comin') (on which the multi-instrumentalist Jones played tuba over Donald "Duck" Dunn's bass line), Soul Man, Who's Making Love, I've Been Loving You Too Long (To Stop Now), and Try a Little Tenderness, among others.
If the button can have the Stooges, I'll say Can (although they did loads of great stuff in the 70s too).
On reflection, maybe Booker T & the MGs. For their solo stuff, but even more for their work as the Stax records "house band."
As wikipedia tells us: -
The Funk Brothers![]()
but unlike The MGs or The Skatalites, they weren't known for their own stuff, they were purely a session band.

....and not enough people have mentioned the Beatles
Beach Boys. Only cos nobody else has said 'em.
I thought about them too. Some blinding tunes, but a bit of a one trick ponySonics?