What if Alexander came to Canada, perhaps to the Calgary area, but moved on before he appeared on any census and moved south into the United States or went directly to the U.S.
I couldn't find him on the 1901 Scotland census but on the 1901 English census there is an
Alexander Davie b. 1880 Scotland, 21, working as a Coachman, domestic servant in the household of William and Sarah Brindle, St. George Hanover Sq. London, London.
1910 U.S. census there is an Alex Davis, 29, single, b. Scotland, both parents born Scotland, is a Partner to the head of the household, occupation miner. This is Thomspson Township, Sanders County, Montana.
Sanders County is west of Flathead Lake and one of the towns within it is Thompson Falls.
1920 U.S. census Alex Davie, 38, b. 1882 Scotland, both parents born Scotland, Immigration year 1904, occupation Homesteader. This is in Smead Township, Sanders County, Montana
1930 U.S. census Alexander Davie, 49, single, b. 1881 Scotland, roomer, Imigration year 1904, occupation Miner in a copper mine. This is in the city of Butte, Silver Bow County, Montana
Death of Alex Davie at 63 b. 1881 d. 11 May 1944 at Deer Lodge County #DL5223
http://geology.com/county-map/montana.shtml - Link to county map of Montana. Deer Lodge is just next to Silver Bow.
Debbie