While it might be helpful to know some HTML, most of the time, an embedded developer (just as an example) will not need that knowledge, not hone it and, as a result, not be able to use it either way. It's the same for other types of software engineers. There are some rather theoretical architects, there are low-level developers (OS, drivers, etc.), there are desktop application developers and last but not least specialized app developers. Web dev is a nice-to-have, but should they know it? I don't think so and I would never require them to.
No, many software engineers don't touch the web. Things like programming embedded microprocessors, building operating systems, writing compilers, data mining, AI, integrating into banks, writing software that controls planes / cars, realtime software, software written in ADA for example that deals with very critical things like manned space travel or even nuclear power stations, robotics, 3D vision applications doesn't require web and even mobile and desktop applications doesn't require web if you're building native applications
Web is a nice-to-have skill, although very in demand skill if you don't have any of the other specialities
Web development is nice-to-have, but not mandatory.