Then you have an ambiguity between names where the Umlaut is replaced with "ae"/"oe"/"ue" (Haendel) and names that are actually written with "ae"/"oe"/"ue" (Goethe)...
But does German have ambiguities here? Yes, it does. In the instrument 'Oboe', the 'oe' is pronounced as two syllables: long 'o' plus schwa. It cannot be written with 'ö'. It should better be written 'Obohe' (my controversial proposal). And dialectally/regionally, 'oe' may be long 'o' instead of the 'ö' sound, like in the city of 'Soest', which could be written 'Sohst', but then people from that city would complain (about tradition, maybe).
ō′bō <-- see? From the American Heritage Dictionary 5th-ed. A schwa is the upside down 'e' which is not present.