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This is not in fact how “all of Hive works” - for example PeakLock stores all of your keys encrypted in local storage (the correct way, which does not store a cookie or send it to the server). Source: the PeakLock dialog:

IMG_9558.jpeg

This is an incredibly negligent way to handle keys.

I am not able to verify this at the moment, but according to the screenshot you provided, when you log in with PeakLock, you provide a pin code. This pin code is stored for XX hours. While this has been done, the key is not encrypted. This mean that while the pin code is 'active', the key is not encrypted.
PeakLock then require a pin code to reactivate (unencrypt) you key after the XX hours.
InLeo does not require a pin code after those hours (may or may not be the same amount as Peakd) but you will be logged out after XX hours, meaning the key will not be in your browser.

Both ways have pros and cons and I don't really think there's a big difference in terms of security