Do all redheads originate from Scotland-Ireland?

No, although Scotland and Ireland have the largest populations of red heard by proportion, in generally red heads would have all mostly originated from Northern Europe (including but not limited to Scotland & Ireland)

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As to why — Red hair — Wikipedia

The genetics of red hair, discovered in 1997, appear to be associated with the melanocortin-1 receptor (MC1R), which is found on chromosome 16. Red hair is associated with fair skin color because low concentrations of eumelanin throughout the body of those with red hair caused by a MC1R mutation can cause both. The lower melanin concentration in skin confers the advantage that a sufficient concentration of important Vitamin D can be produced under low light conditions. However, when UV-radiation is strong (as in regions close to the equator) the lower concentration of melanin leads to several medical disadvantages, such as a higher risk of skin cancer.

The MC1R recessive variant gene that gives people red hair generally results in skin that is unable to tan. Because of the natural tanning reaction to the sun’s ultraviolet light and high amounts of pheomelanin in the skin, frecklesare a common but not all-inclusive feature of red-haired people. Eighty percent of redheads have an MC1R gene variant

Red hair is the rarest natural hair color in humans. The non-tanning skin associated with red hair may have been advantageous in far-northern climates where sunlight is scarce. Studies by Bodmer and Cavalli-Sforza(1976) hypothesized that lighter skin pigmentation prevents ricketsin colder climates by encouraging higher levels of vitamin D production and also allows the individual to retain heat better than someone with darker skin.

In 2000, Harding et al. concluded that red hair is not the result of positive selection but of a lack of negative selection. In Africa, for example, red hair is selected against because high levels of sun harm untanned skin. However, in Northern Europe this does not happen, so redheads can become more common through genetic drift

Studies have shown that the percentage of Irish residents with red hair is indeed around 10 per cent. So the Island of Ireland would appear to have more red heads than Scotland and Wales.

From 13 fascinating facts about redheads that everyone needs to know

LESS THAN 2% OF THE WORLD’S POPULATION HAVE RED HAIR

That’s approximately 140 million people. Scotland boasts the highest percentage of natural redheads, with 13% (40% might carry the gene there) while Ireland comes in second with 10%.

RED HAIR CAN OCCUR IN ANY ETHNICITY

It occurs more frequently (2–6%) in people of northern or western European ancestry, and less frequently in other populations.

Though more usefully from Mapped: Which countries have the most redheads?

The Celtic nations lead the way. According to the map, around 10 per cent of the population of Scotland, Ireland and Wales boast a carrot top. Though the data used to produce the map isn’t clear, a trip north of the border, or across the Irish Sea, would appear to back up its claims. Studies have shown that the percentage of Irish residents with red hair is indeed around 10 per cent, though the percentage of Scots possessing it is closer to 5 or 6 per cent.

Red hair is relatively common in England, Iceland and Norway, too, while Germany, Sweden, Finland, The Netherlands and northern France are all ahead of the global average.

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