Spellcheck is surprisingly influential and can even cause language change. From Gretchen McCulloch's book Because Internet (p. 67):
Default computer spellings are powerful enough to have created a shift in British English since the 1990s: while American English prefers a Z in words like "organize" and "realize," British English has traditionally used both -ise and -ize spellings. But spellchecks have tried to prevent people from spelling the same word differently within the same document by enforcing "organise" and "realise" all the time when set to British English, leading to an upswing in -ise endings among the general British typing public and the perception that -ize is only for Americans.
Anecdotally, I sometimes use a website called LanguageTool to "proofread" important texts, and it's good enough to point out simple mistakes that I can no longer see after working on a text for too long. Every once in a while, though, (and especially in languages other than English) it incorrectly marks things as wrong, but it has this weird kind of authority (it's a spellcheck, after all!) that makes me question myself.