The tales of the Jedi were spread far and wide around the galaxy in the pre-Empire period and even those unaware of them could be easily impressed by the display of their awesome powers. Since the Jedi were aligned with The Republic, it had a strategic military advantage over any other political entity. Did The Republic ever used this advantage to dominate in negotiations? Perhaps by threatening to unleash their might onto the opponent?
Yes. Many times.
The Republic was established in 25,053 BBY. The Jedi operated as its military (or at least a branch of it) from the beginning until 1000 BBY. So they worked as both an implicit and explicit threat for that whole period.
After the Jedi were demilitarized following the Seventh Battle of Ruusan, they were still the most individually powerful beings in the Galaxy, and since they served the Republic, the Republic could still use them as an implicit threat.
An example of this is when the were sent on a diplomatic mission to the Trade Federation during the Blockade of Naboo.
Hoping to resolve the matter with a blockade of deadly battleships, the greedy Trade Federation has stopped all shipping to the small planet of Naboo.
While the congress of the Republic endlessly debates this alarming chain of events, the Supreme Chancellor has secretly dispatched two Jedi Knights, the guardians of peace and justice in the galaxy, to settle the conflict....
The Phantom Menace - Opening Crawl
The message here was "Stop it, or else." Otherwise, the Chancellor would have sent a normal diplomat. Guardians of peace and justice in the galaxy they may be, but they still have an enormous amount of inherent power and the authority to wield it (within the confines of the Jedi Code) to achieve their goals.