The second option gives you a deterministic number of opponents the player will face (within a range); as in 0-1 in your case. In the case of Random Encounters, one player may hit as many as 5 if you don't cap it; while they would likely have a smaller chance of hitting 0.
The main benefits I can see is it allows you to:
Have cool items that maybe temporarily reveal where the thing was or is, giving depth of gameplay
It can be used on particularly tough opponents, encouraging the player to find those items, but they don't have to. This allows the player to make a meaningful decision: "Should I search for items X or Y or should I gear up for a massive fight?"
Having it "hunt" the player a bit, maybe the player "makes noise" when they move or similar, allowing further depth of gameplay where moving slower, skirting the edge, etc. allows you to dodge it with better; possibly with a "throwing rocks" mechanic that Far Cry and similar games had
I can't think of anything outside of these types of mind-games and decisions that it allow. Remember that Games are a 'Series of Interesting Decisions', so when implementing a thing; see if it affects the gameplay in a way that affects decision making. If not, you can consider removing it; simplifying it; or adding decision making to make it more interesting.
The above Lures, Monster-Finders, etc.. may all be single use things; allowing you to skip one of a number of fights; making which ones you do meaningful. Additionally, being "hunted" and having things assist you with that adds a lot of atmosphere, where random encounters are just there for grinding and simplification of development.