Spellcheck in Overleaf is very basic. It is nothing compared to dedicated tools such as Grammarly. I used to copy and paste into Docs back and forward until recently. I created this chrome extension that copies the content of the file visible in the editor; filters the commands/keywords and pastes the plain text in a textarea (Right area). This textarea is displayed over the pdf preview. Extensions such as Grammarly can't check complex editors such as Overleaf, but only work in textareas. Changes made to the textarea, either by extensions such as Grammarly or yourself, are evaluated and changes are merged back to the editor.