Actually, A: is not valid with the default compose bindings (it is however valid with vim digraphs). In general umlauts or trémas are inserted with the double quote (A").

Worth noting that international layouts (with dead keys) support many of those without the use of compose key (for obvious reasons not your first example – but the € symbol has its own key combination at least where it's commonly used). I admit it might be hard to adjust to caret being a dead key, though.

Dead keys support accents, and a few symbols printed on the keyboard are discoverable, but I find the Compose key is much more intuitive for occasional use.

I don't yet speak the language of my adopted country, so it's better for me to keep []{} etc where I like them in the British layout, and use three keypresses for typing the ø in a (place) name like København.

If I do end up typing lots of Danish, I'll probably map AltGr+A,E,O to Å, Æ, Ø. É is rare, so I'll still use the Compose key for that and German / Swedish names.

I use the default German keyboard layout. Even with dead keys I wouldn't have Nordic characters or the Turkish dotless i and dotted I. I love the Compose key. It's actually making me consider using the US layout because the lack of German glyphs was what was keeping me back.