The first observation does not hold for all groups: Ono (1963) found examples where the Tamagawa numbers are not integers.
The second observation, that the Tamagawa numbers of simply connected semisimple groups seem to be 1, became known as the Weil conjecture.
V. I. Chernousov (1989) removed this restriction, by proving the Hasse principle for the resistant E8 case (see strong approximation in algebraic groups), thus completing the proof of Weil's conjecture.
In 2011, Jacob Lurie and Dennis Gaitsgory announced a proof of the conjecture for algebraic groups over function fields over finite fields,[1] formally published in Gaitsgory & Lurie (2019), and a future proof using a version of the Grothendieck-Lefschetz trace formula will be published in a second volume.
Ono (1965) used the Weil conjecture to calculate the Tamagawa numbers of all semisimple algebraic groups.