It's not exactly right to say that a/å/ä and o/ö have absolutely no relation in Swedish. There are plenty of language cases where they act more like umlauts, such as stor/större (large/larger) or få/färre (few/fewer). Despite this, they are still always treated as separate letters.

Also, the history of the letter shapes are the same as the german umlauts, with e being written above a or o before writing ä or ö and o written above a before writing å.