Should I cut this friend off?

My friend is from out of state and doesn’t have any friends here yet. We met through work and we got along really quickly and have gotten really close and hung out a few times.
There’s this other girl who just started the job and she’s around our same age. She’s a little quiet but she’s cool and my friend and I have tried to include her in a few hangouts and we’ve all gotten close. There’s been a couple of times where this new girl sees how close my friend and I are and will feel awkward and has told me that she feels bad because me and my friend met sooner and are already close but we all still hangout sometimes.
Lately my friend keeps saying how she loves the three of us hanging out and will invite us to stuff but I get the feeling that she only has fun when it’s all three of us and not just the two of us anymore. Like when the other girl cancels, my friend seems less enthusiastic for us to hangout alone and won’t take the lead on scheduling plans even if she mentioned it before. Maybe it’s all in my head I don't know.
So she’s having a birthday dinner and she told me and the other girl about it like two weeks ago. She made it clear she was excited and wanted BOTH of us to be there since she doesn’t have many friends here yet and we’re her only two friend here.
The other girl said she probably wouldn’t be able to come and my friend was convinced I was still coming. Yesterday when all three of us hung out the other girl said she might be able to make it after all and I wanted her to come so that I wouldn’t be the only guest.
Today the other girl texted me saying she left something in my car but said she would get it tomorrow so I assumed that meant she wasn’t coming to my friends party after all. My friend hasn’t said anything about me coming all day and neither have I and I’m hurt because I feel like she only cares when the other girl says she’s coming too. The time for the party has already came and passed and neither of us said anything. Who’s wrong?
Should I cut this friend off?
Post Opinion