Yes, travel insurance is essential for anyone flying into Kilimanjaro Airport to climb the mountain, and here the honest core of the answer is altitude rather than anything else: Kilimanjaro's summit sits near 5,895 meters, and acute altitude sickness is a genuine, common risk on nearly every route up, regardless of fitness level. Even travelers only passing through for a safari elsewhere in northern Tanzania benefit from the same baseline medical and evacuation cover, since the airport itself sits well outside any major hospital city.
Moshi and Arusha, the two towns nearest the mountain, have hospitals that handle routine treatment and can manage most altitude-related cases brought down off the mountain, but anything more serious typically means further evacuation, when needed, to Nairobi or another major hospital abroad. Confirm your policy explicitly covers high-altitude trekking and mountain evacuation rather than only standard illness and injury, since altitude sickness is treated as its own risk category by many insurers.
Reputable trekking operators build acclimatization days into the itinerary and watch climbers for symptoms throughout the ascent, but insurance still needs to cover the descent-and-evacuation scenario directly, including cases severe enough to require porters or a stretcher team down one of the standard routes. Lower-altitude areas around Moshi and Arusha also carry a general malaria risk, so check that your policy and your own preparations account for that alongside the mountain-specific cover, rather than treating altitude as the only concern worth insuring against.
Tanzania's long rains, roughly March through May, and shorter rains around November, can affect trail conditions and visibility on the mountain and are worth checking against your travel dates, since some operators reschedule or reroute climbs around the wettest weeks. Keep any guide or clinic paperwork from the trek, along with receipts for evacuation or treatment, since claims involving a multi-day trek rely heavily on documentation from the operator as well as any facility you reach afterward.