I decided to look within Dolphin builds, if there would be any hints of what broke the OpenGL backend for Intel HD. I eventually found out, that Dolphin revision 4.0-583 was the one that breaks the backend with results described above.
The revision was the one that introduced buffer_storage (OpenGL 4.4 extension), so this suggests that Intel HD might be handling this incorrectly atm.
OpenGL backend works on AMD and NVIDIA hardware, so the buffer storage code shouldn't be wrong in the software.