No when the battery is really completely empty the phone will not start when plugged in.
Also this does not work with the FP3
@Ellen_Lees it seems you are hit by different issues, for the charging issue I think its better to follow up in one of the many topics around this
https://forum.fairphone.com/search?context=topic&context_id=90386&q=FP3%20not%20charging%20order%3Alatest&skip_context=true