I think your missing the key point about the dates. The census dates are irrelevant as far as the surgeon is concerned, the 1st entry from the London Gazette I quoted means that Alexander Sinclair the doctor, was already a qualified surgeon and serving in the army in September 1838 and was still serving in 1855, so the other Alexander the type founder must me a different man.
If you check other Gazette issues you may find a listing of his demise and check Edinburgh papers perhaps you’ll find an obituary for the surgeon. Not sure where Scottish Wills are held but you should also check those out too.
Re wars, for the time frame you’re looking at 1851-1861, in addition to the Crimea war 1853-1856, there were also: North West Frontier 1849-58 & 1859-78; 2nd Burma War 1852-53; Persian War 1856-57; Taiping Rebellion 1856-66; Indian Sepoy Rebellion [Indian Mutiny] 1857-58; 2nd China War 1857-62 and 2nd Maori War 1860-61.
In addition the British Army had posts all around the world and in the West Indies, Middle East, Africa, East Indies and China, more men died of disease than in battle.