So one would actually have to remember to call the country TALT-129kyie?

You can type an umlaut (over a u) on a US-International QWERTY keyboard by typing a double quote followed by a u.

And if you are using Windows with, say, US-English QWERTY as your default keyboard layout, its easy to add US-International and switch between them with ctrl-shift; they have the same layout (so your keycaps are still write), but some of the punctuation becomes dead keys that can function either for the punctuation or diacriticals depending on the following keypress.

Because the relationship between the punctuation and the diacriticals they work for is mostly visually intuitive, I find it a lot easier than memorizing Alt-key codes. Especially when using keyboards that don’t have a separate numeric keyboard in the first place like small laptops, and big laptops with relatively powerful dGPUs (the former because of total space available, the latter because they use more space for venting).

On Macs, you can long-press the "u" key and you'll see options for û ü ù ú ū. There's a number underneath, so just long press "u" and then type the number for the character you want.