Q: Do sailors today regularly rely on any tools that sailors two hundred years ago used?

Sure, how about:

• Crew knife
• Ship's compass
• Hand bearing compass
• Telescope/binoculars
• Marlinspike
• Sextant (still required to earn the Offshore Passagemaking certification from the American Sailing Association)
• Chronometer/ship's clock
• Charting tools (parallel rules, dividers, pencil)
• Nautical charts
• Depth sounder (to make the nautical chart)
• Boat hook
• Telltales and sail ticklers (little bits of yarn to indicate wind direction and flow around the sails)
• Hand pump
• Bailing bucket
• Wind vane (masthead fly)
• Anchor
• Oar
• "Tools" used to hold things to things on the boat: blocks and tackles, shackles, cleats, winches, turnbuckles, padeyes, cotter pins, sheaves
• Cannon. Unfortunately few sailors today really know how to fire their ship's cannon properly.