Indeed Sublime itself doesn’t do any sort of error checking at all, except minimally in the form of syntax highlighting showing you something is structurally broken somehow (for example you might detect that a tag is not closed because everything is the wrong color).
SublimeLinter itself also doesn’t do any sort of error checking either, it’s just a framework that error checking plugins can use to display their results. Essentially it just does all of the heavy lifting and work of interfacing with Sublime so that external tools can concentrate on error checking.
There appears to be a few HTML linters available for SublimeLinter on Package Control, including some HTML tidiers.