The title screen has a lot of charm. Wendy drawn in a red hat with a revolver, two silhouetted enemies (a huge brown one and a skinny blue one with red eyes) in front of a purple stone building, warm desert colours and a western font. Very cohesive. The in-game HUD is clean too, three hearts, an empty item slot outlined in blue, a reload bar, and big SCORE and TIME in bright blue at the top. The desert with stone fences, barrels, cacti and a cow skull has consistent art direction.
Real credit for being honest in the description, flagging up front that the build was rushed and that the high score and lives don't reset without reloading the page. I'd take that over discovering it myself any day.
The obvious priority: the START GAME button doesn't respond to clicks. I tried single, double, on the text and around it. It only worked when I pressed Space/Enter. Since the screen doesn't say "press space", anyone who only clicks will assume it's frozen. That one fix probably rescues a lot of votes.
The bug you flagged (score and lives not resetting) stings most in a high score game specifically, because the second run starts dirty. And the timer, which starts around 51 and drops fast, is running while you're still working out the map. I walked a fair bit and finished with SCORE 0 because I couldn't find an enemy near the spawn, and the map is big and empty at the edges with no indication of where to go.