Here's my personal take on the subject, it was too long to make a comment.
First, I would recommend reading my questions on Sanhedrin, especially "evaluating-sages'-wisdom-when-electing-for-the-Sanhedrin".
The question of capital punishments can be reduced to Sanhedrin's authority to take people's lives, and here we can ask where does this authority come from. In fact, the Torah prescribes the punishment but never details how the Sanhedrin should be elected, beyond Moses' elders, that were chosen by his Holy Spirit. The Mishnaic proposition (Sanhedrin 4,4) of "everyone (simply) knows his relative position" without the need for elections seems somewhat problematic.
In my understanding, (without actually explaining how it worked in fact) there's no way to obtain judicial authority in the eyes of the people, but by adjoining the Sanhedrin to the Temple, and claiming [some] Holy spirit "intervention" with the judges, as the verse says "כִּי מִצִּיּוֹן תֵּצֵא תוֹרָה וּדְבַר־ה' מִירוּשָׁלִָם" - the closeness to the Temple gives Sanhedrin's decrees and sanctions the necessitated divinity. (See also Rashi on "אלה המשפטים" - "ולמה נסמכה פרשת דינין לפרשת מזבח, לומר לך שתשים סנהדרין אצל המזבח".)
Naturally, when the Temple was destroyed and the Sanhedrin lost its associated divinity, it also completely lost its authority in the eyes of the people. This can clearly be seen throughout the Talmud - rabbis themselves never appeal to the Sanhedrin, never even mention a functioning one.
Indeed, without the authority of the Roman government, without national elections, or a prophet, or a divine revelation, why would people trust a bunch of rabbis who can't agree between themselves even on how the Sanhedrin should operate? (see how-sanhedrin-was-possible-without-agreement-on-procedures".)
This seems to be the real (down-to-earth) explanation, why capital and corporal punishment at large was abandoned right after the destruction of the Temple.