I got something similar to Hilary in Google docs, a red underline under a word with correct spelling that had " at the end. When I clicked to see what the issue was, it told me there were no suggestions but the red underline stayed
Almost as if your browser dictionary is configured wrong Brian, which is probably not a bad approach route to take at all.
I would simply be not taxing my noggin, but rather exploring ways a user might find the spell checker an annoyance; finding ways that it might miss-configure; and find arguments to support red squiggles being a poor colour choice.
At first thought, it seems like testing a spell checker might be easier to demonstrate that it has problems, rather than to show its ‘correctness’. There are just so many different things that would need to be tried with a spell checker, including…
Basic word spelling
Grammar
Recognition of pronouns
How does it know someone didn’t name their kid Aberaham Lyncoln?
If it correctly recognizes misspellings, what does it suggest instead? (This editor originally flagged misspellings, but now it doesn’t. What’s up with that?)
Does it ignore strings of numbers and/or characters, like 08/31/2020 or (2+2)*3=12?
Does it care about extra spaces in sentences?
Who about suggesting How when typing who?
What about it’s detection of incorrect use of its vs. it’s?
Are abbreviations OK?
Aside from targeted checks like the above, I might try pasting in large chunks of text (even from public domain books) to see what gets flagged - and compare against something like Word. A flawed baseline is still a baseline, right?