“I really cannot understand why the whole world bases its 24 hour clock timing on the sun being directly overhead at 12 noon.”
Because, Hymie, (making some minor allowances for the unevenness of the earth’s orbit and axial tilt) that’s precisely what the definition of “noon” is !!
Standard time across the UK was set up as recently as 1847. Until then each town or village maintained its own version of “noon” which varied by about 30 minutes from east to west. But the coming of the railways meant a standard time was needed and “Railway Time”(based on Greenwich Mean Time) was adopted.
Because the earth is split into time zones for convenience the time shown on the clock will not coincide precisely with the Sun’s highest point in the sky but instead will vary a little either side. Each time zone covers 15 degrees and so ideally they should end 7.5 degrees either side of the line of longitude on which they are based. But of course the time zones usually relate to a group of countries so as to make matters more convenient. This works fine for the UK and Ireland as they lie roughly between the longitudes nine degrees west to two degrees east of zero degrees longitude. British Summer Time is really appropriate for longitudes 7.5 degrees east to 22.5 degrees east – that is none of the UK at all. (As an aside, the time zone which is one hour ahead of GMT is also inappropriate for all of Spain, Portugal, Belgium, Luxemburg, Holland and all but a tiny bit of eastern France. All these countries lie west of Longitude 7.5 degrees east).
Your Caribbean adventure was not a fair comparison. All tropical locations get near enough twelve hours of day and night throughout the year. In parts of the UK there can be almost nineteen hours of daylight in midsummer and nobody can realistically take advantage of the whole lot, whatever time of day you call it when the sun rises.
I believe the reason that BST covers seven months of the year is that its originator, William Willett, saw the idea as “Daylight Saving Time” (a misnomer if ever there was one) and the period April to October was the maximum that would be tolerated in order to get the bill through Parliament.
My own view is that really is no point messing about with the clocks twice a year and that the UK should remain on GMT throughout the year. It cannot “save daylight” and the argument that accident numbers are reduced by lighter evenings (at the expense of darker mornings) has never been substantiated. The last time “GMT+1” was tried coincided with the introduction of the drink driving laws so a comparison was not all that useful.