My experience has been that this may not be possible, but probably depends on which phone you're using. My current and past two phones flat out would not allow me to set SwiftKey as the global spell-checker. (I can autocorrect with it, but with two languages plus dialect plus slang in the "learned" dictionary... this is not especially useful.)
I would check that your default spell-checker is actually, or even can be, set to SwiftKey. On my current Samsung with android 11, the spell-checker setting is completely integrated and only available under "Samsung keyboard settings" (meaning I can only use the native Samsung keyboard if I want to enable it), while on the older Huaweis (android 6 & 9), it was called "Spelling correction," but the only selection possible was Gboard.
(Someone elsewhere on SO suggested this was a change in android 5, and looking at the releases, the SpellCheckerService class pops up in API 14/android 4, so it may be that the global function didn't even exist until then.)
Edited to reiterate: N.B. that I'm specifically responding to the question as to "spell-checker" and not "autocorrect" (as it sounds like @metshein is referencing below), which (somewhat confusingly) are actually two different settings. The OP mentioned "red lines" below words, which is a spell-checking function separate from autocorrect, which can be enabled or not within SwiftKey.