Korcula In October

I have not been in October, so take it with a grain of salt, but given the overcrowding in summer, I would think early October would be a good time to visit, as long as the weather holds. People do live there, so I think enough restaurants will be open. The boat tour is actually the thing I would be most worried about, so if you have that sorted already, I personally would take the risk on it being quiet--I'd be 1000X more enthused about an October visit versus August for example.
You could call around to some of the mainland wineries and see if they are any different--it could be that that is when the season ends overall. There may be some tasting opportunities even if the tours are over.
Although they obviously have an interest in being positive, maybe an email to the tourism office would be encouraging?
https://visitkorcula.eu/en/visit-korcula-english/

I was in Korcula toward the end of September, then in Hvar until October 5.

By October 5 the weather began changing with some of the Bura winds blowing in and rain falling.

Over the years, I’ve usually been in Europe in September and at the beginning of fall. And I have noticed—by the first week of October—there have often been big weather changes toward rainy, windy days In Germany, in Switzerland, in France, in Greece and in Croatia.

I think it’s a pattern you need to be ready for.
I’ve been told that once the windy, rainy days arrive, the beautiful, warm beach, biking & hiking weather is mostly over.

If the tour or restaurants are owned by locals they will be open. Some places are owned and operated by people who do not live on the island. When they feel like returning home, they close up.

Can't help you with things being closed or not. I would presume that since it's primarily a tourist destination entering the quiet time of the year, you might be correct in your concern. Im easily entertained, so it would wreck things for me if half of everything were closed.

As for the weather. Weatherspark says, on average, it's pretty cool (56 to 66). But it's neither the high season for high winds nor for a lot of rain in quantity of rain or quantity of rain days. Given low temps and a constant breeze it might not be as fun as if you had visited a month sooner.