The fact that they spend their precious time to look for you on social media is a positive sign.
Does having a Linkedin with no connections reflect badly on me?
YES, If you are a professional with N years of working experience. This might indicate you don't have a good social and communication skills.
However, for your case it's perfectly fine, as you are a fresh graduate and you just created your account couple of month ago. They are not expecting to see C level people in your connections.
Am I overthinking this?
Yes, you are. Instead, give them a call (Not an email) a week after you apply, and follow up with them. This shows your interest and enthusiasm.