Pilot reported a lack of NOTAMs regarding ongoing construction and intermittent airport closures for MYCA airport.

Narrative

Yesterday I flew to MYCB on Cat Island; Bahamas. FAA notam website has shown MYCB to have been closed recently for a day or two and up to a week or so this month; due to runway repaving there. Prior to my flight; I checked and no NOTAMs were listed. My trip went fine; runway mostly repaved. Today driving on the island; I stopped into MYCA; Arthur's Town; also on Cat Island. This airport has men and equipment on the runway in a massive construction operation; single runway; no taxiways. No NOTAMs are listed on the FAA site; which suggests the airport is usable. Since the MYCB NOTAMs have been updated regularly; it seems reasonable to assume MYCA would be also. I called 800wxbrief to report this; but they said they have nothing to do with Bahamas NOTAMs. The trouble is their briefers are saying MYCA is open; but it definitely is not. Pilots could be dangerously led astray here. I asked at least for a PIREP to be filed; and one briefer said he'd add that. I will call Nassau when I return to the US next in the meantime; this is a potentially unsafe situation. I myself considered MYCA as an alternate on this flight. US pilots need reliable NOTAMs everywhere. Not all countries are as attentive to posting information; but that's a separate problem. Today; I notified FSS of an airport closure with men and equipment on the runway; and they say they can't/won't make that information available to future inbound pilots. That's not a Bahamian problem; that's a US problem. FSS is now aware of a hazard; yet they brief the opposite; that it's fine to try to land at MYCA when it's definitely not. A notam in the US system should be added immediately if only to say 'Pilot reported airport closed with men on runway' or something similar.