I used to use "pass" like others here, but did not like the Android experience.
*edited to add: and we use the 1Password team account at my day job -- and are satisfied customers. I'm sure other products work well too -- just my one data point.
I believe the browser addons do not share this shortcoming, though.
LastPass, on the other hand, is in a different category. It _claims_ to have full Linux support, and for a long time they did, but more recently -- as you point out -- copy/paste in their browser plugins stopped working properly when the binary component of the plugin is enabled on Linux. Since the binary plugin component is required to work with attachments, Linux users have been forced to choose between working copy/paste and the ability to manipulate attachments. They've known about this bug for many months and have not fixed it. In fact, this is one of the unfixed bugs which drove us to finally evaluate alternatives to LastPass.
Turns out, the hardest thing is not the crypto or the browser to app communication, but parsing the HTML to accurately find the login forms. If I remember correctly, the browser plugin on Mac actually sends the entire HTML to the app for parsing. The parser is probably quite complicated and they avoid reimplementing it in extensions that way.
[1] https://medium.com/@paulsc/making-a-1password-client-15dd39a...
Also, great post. I love reading reverse engineering stories.