I'd point out that the title of your question states "Do the Jedi..." while the body states "Why do the Jedi...". Not sure the actual question here, but here is my take:
It is not true that the Jedi always kill Sith on sight. There are several noticeable instances of Jedi attempting to, or actually capturing, Sith.
Capture of Darth Revan: During the Jedi Civil War, Jedi agents successfully captured Darth Revan, though at the cost of most of their team. Source: Knights of the Old Republic (video game)
Capture of Count Dooku: During the Clone Wars, Anakin and Obi-Wan capture Count Dooku from a band of pirates. Source: The Clone Wars (TV Show)
Capture of the Dread Masters: During the Great Galactic War, a Jedi strike team was successful in capturing the powerful six Dread Masters, and sequestering them away in a maximum-security prison. The purpose was the Jedi council believed the masters were too valuable. Source: The Old Republic (video game)
Capture of the Sith Emperor: During the Cold War, a strike team sought to capture an redeem the Sith Emperor; however, they were unsuccessful and were themselves turned into Sith. Source: The Old Republic (video game)
Capture of Darth Sidious: At the end of the Clone Wars, a group of four Jedi Masters confront Palpatine upon learning he is a Sith Lord, and seek to arrest him. However, they fail and are all killed. Source: Revenge of the Sith (film)
Capture of Tahiri Velia: At the end of the Galactic Civil War, Tahiri was captured by Ben Skywalker after a short duel. Though it does not appear Ben had specific plans - whether to kill or capture before the duel, he saw an opportunity and took it when it came up during the duel. It should be noted that Tahiri was at this point a very disenfranchised Sith apprentice, and not one very deep in the Sith way. Source: Legacy of the Force - Invincible (novel)
So you see, there are multiple cases, in varying media, including one of the main films, and from multiple time periods, that show instances where the Jedi attempt or are successful in capturing Sith. In some cases, it is a very powerful Sith that they attempt to capture.
In the films, we see many instances where the fight is just 1 or 2 Jedi against a Sith, and often, not planned encounters. It would be difficult to attempt to capture a Sith when you do not have the benefit of surprise, and without the benefit of numbers. There was really no hope for Qui-Gon and Obi-Wan to capture Maul, or for Obi-Wan and Anakin to capture Dooku in Episode II. While Anakin does overpower Dooku in Episode III, the plan did appear to capture him; however, Anakin did violate the Jedi code by killing him when it wasn't necessary. Of course all the encounters in the original trilogy are instances where the Jedi was the weaker combatant or had other extenuating circumstances precluding a capture attempt. So it is a fight to the death usually because the Sith would grant no quarter, not because the Jedi will not.