RefRunner gives your list a link to a live .bib file that Overleaf reads, so a new reference is one click from your paper's bibliography. Works on Overleaf's free plan.
Overleaf's free plan can't link to Zotero or Mendeley. No problem. RefRunner will give you a URL that Overleaf will pull from. Start with your references in RefRunner. They can come from a paper, a list you paste, a bib file, or any of RefRunners search tool's. Save the list. Then choose File, Overleaf link.
Click "Make link," and copy it. Anyone with this link can read the reference list, so be careful where you share it. Don't worry; you can easily invalidate it and get a new one. In Overleaf, open your project. This short paper makes three citations from a test bibliography. From the File menu, choose "New file," then click "From external URL."
Paste the link, and give the file a name. I used refs.bib. Next click "Create." There they are. Your references, with their DOIs and page numbers, ready to cite in your document. Update the bibliography line to use "refs," and recompile. The references now come from RefRunner.
Now you need to add one more reference. Back in RefRunner, find and add it. The list saves itself. In Overleaf, click Refresh on the file, and recompile. The new reference is there. Search in RefRunner, click Refresh in Overleaf. That's a smooth workflow.