I deleted my bitcoin-qt thread when I was reminded it downloads the entire blockchain, but how about Electrum?
It has the blockchain stored on the internet, while it does not store your actual wallet there.
New: Kanri (Oct 9, '24), Platform 29.5.3 (Jun 27, '24)
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Electrum seems like a decent fit, so I packaged up both of them. Let me know if there are any issues.