Yes, travel insurance is worth carrying for a trip through Edinburgh, and largely because of what comes after the airport rather than the city itself. Edinburgh has good local hospital care by any standard, but many visitors use it as a gateway onward into the Scottish Highlands, and that onward leg brings a genuinely different risk profile - remote roads, changeable weather, and winter conditions a routine city policy may not anticipate. Even a short city-only stay benefits from ordinary trip-delay and medical cover, given how easily Scottish weather disrupts flight schedules at the airport itself.
Whether your own health cover applies in the UK depends entirely on your country of residence and plan, and this page cannot answer that for you - check whether your existing cover extends to the UK specifically before you travel, since the rules differ for visitors from inside and outside Europe. A standalone travel policy is the simpler, more reliable option for most visitors regardless of what a home plan might technically include.
Hillwalking and Munro-bagging in the Highlands and the Cairngorms are genuine draws for visitors flying into Edinburgh, and both carry real risk from fast-changing weather and remote terrain with limited phone signal. Winter, roughly November through March, adds snow, ice, and storm-related flight and road disruption on top of that, including closures on Highland routes; confirm your policy covers hillwalking and winter travel delay before heading north. Trekking poles and proper gear reduce risk but do not replace a policy that actually covers mountain rescue if something goes wrong far from a road.
Keep any clinic paperwork, mountain rescue or police incident reports, and receipts for delayed flights or unplanned accommodation if Highland weather disrupts your plans. Many policies treat weather-related flight or road disruption as a standard delay claim, so check how your policy defines it and what documentation it wants before you assume reimbursement happens automatically.