Lighthouses are not cheap, quick, or easy to build or maintain. Because of this they are only likely to be situated where they're going to give the most benefit
So what benefit does a lighthouse give? It's a big light, that can be seen at a great distance. You can (and usually do) also have shutters rotating around to make it appear to flash in different patterns, and have filters to make it appear different colours in certain directions
The pattern of flashing allows you to give each lighthouse an identifiable pattern so that the person at sea can work out which lighthouse they're looking at
Some of these patterns have specific interpretations. I.e. 2 quick flashes every 5 or 10 seconds is an isolated danger mark, often used to mark small rocks or submerged wrecks. If you see one of these, you just don't go near it and you're good. Another common pattern is that of a cardinal mark which tells you to stay a particular side of the light. As your lighthouse is on the Eastern tip of your island, it would likely be an East Cardinal Mark, which would flash bursts of 3 flashes, warning people to stay East of it (North Cardinals flash continuously, South Cardinals flash 6 times possibly followed by a long flash, and West Cardinals flash 9 times)
If none of those apply, you can still take a bearing from the lighthouse to try and get a fix on your location. You get your compass, measure the bearing to the lighthouse and use this to draw a line on your chart that you know you're on, with another light you can get an approximate position (more lights will generally give you a better and better estimate). Knowing your position is always useful, and historically was a necessity for navigating narrow channels after dark (nowadays these channels would have electrically lit buoys marking the edges of the channel making regular fixes less necessary)
I also mentioned that you can add filters to make the light a different colour in certain directions. This is important because getting an accurate bearing at sea is tricky. Your boat may be rocking or rolling, or being smacked by waves making it hard to keep your compass steady, and if there's a lot of iron onboard the ship (obviously less of an issue historically than with modern steel ships) this can cause magnetic deviation (where the iron onboard the ship bends the magnetic field lines resulting in compasses not pointing as expected) which requires careful calibration to properly account for, and in general you also need to account for magnetic declination (the fact that whilst charts are made with respect to true north, the compass reads magnetic north, which also moves) which requires a fairly easy calculation to account for, but one which people will inevitably occasionally mess up
So, if you have a narrow channel, or harbour entrance that you want people to be able to follow, you can arrange your light at the end of the channel or entrance, with the filters set up so that it appears white if safely in the channel, red if too far to port (left) as facing the light, or green if too far starboard (right) as facing the light. As these filters are fixed on land, they can be set up very precisely and their usefulness won't be as affected by strong weather or local magnetism on the ship
Another common technique here (especially for particularly narrow channels), is to use transits, where there are two lights set up, a low one at the end of the channel, and a higher on further inland. These are set up so that, whilst on the channel the higher light appears directly above the low one, but they'll be out of alignment if you're not in the channel. If you're too far to port (left) as facing the lights the top light will be to the lower light's left, and if too far starboard (right) as facing the light the top light will appear to the lower light's right (colours are reversed I believe in the Americas and Japan, because they're buoyed focussing on leaving ports rather than entering them)
This is just a quick rundown of how navigation lights work, but hopefully gives you an idea of why they're not just for putting on windswept rocks, and your location is a perfectly plausible location for a lighthouse (unlike for instance an straight section of deserted coastline)
The patterns I've listed here are the modern standards regulated by the International Association of Marine Aids to Navigation and Lighthouse Authorities (IALA, after the previous name which left out the middle bit) which was founded in 1957. Obviously your world wouldn't be quite the same, but in an area with substantial hazards or shipping, there would likely be equivalents of the early Lighthouse Authorities (e.g. Trinity House, founded in 1514 originally for managing the Thames but now covering all of England and Wales, the Northern Lighthouse Board, founded in 1786 managing Scotland, even though at the time it only had a handful of lights, and the Commissioners of Irish Lights, also founded in 1786 and managing all of Ireland, and doubtless examples from outside the UK)
In particular, whilst the categories of mark are likely to transfer fairly well (ones for marking isolated dangers, dangers to stay a particular side of, general location, and sectors are fairly natural once you've developed the technology for such flashing lights and colours), the mnemonics the patterns reference will likely be different. The flashes of cardinal marks in our world references a clock face, so in your world, if clocks are different, so might the cardinal marks be. Or you might use a different mnemonic altogether (if your culture associates colours with cardinal directions, your cardinal marks might be coloured to match). Likewise we have a consistent port (left) is red, and starboard (right) is green colour scheme used in sector lights, lateral marks of channels, and lights on individual ships. In your world, to the extent this is standardised (which would likely not be a huge amount), the colour scheme is likely to be different
