Turns out there is a nice project that provides another way of accessing some parts of the Web that doesn't require using a browser. It's called Weboob.
The abbreviation stands for "Web Outside of Browsers". It's written in python and provides a bunch of modules that behind the scenes interact with various sites either through API or just by parsing site contents in custom ways.
In addition to providing a GUI, it also enables use of well-known CLI tools with data from sites Weboob knows about. Some things are available both in GUI and CLI forms.
It has programs for checking weather, searching torrents, fetching lyrics and others.
However, set of supported services will not necessarily satisfy your particular needs. It would be nice to have even more back-ends available.
In the early decades of the Internet it was more heterogeneous place than it is today when even things that aren't transferred over HTTP are still viewed inside browsers. Projects like Weboob could relieve users from the need of using inconvenient Web-interfaces and provide more variety to the ways users interact with WWW.