I think you're exactly right. Go back to Chapter 11, read the conversation MC has with Natsuki, and it makes sense.I am going to call it here, she went with one of those, if pushing doesn't work pull approaches. She treated the MC coldly and then either hired someone or had a friend pass for another guy to try and put a fire under the mc. The MC then was a lot more sensible than most MCs in mangas and had none of it broke it up right then and there.
Things didn't go to plan, and most likely the plan was pushed by Natsuki(the girl in glasses that invites the MC to the party his ex is at on chap 16) she also mentions the MC was a longtime friend of ayaka(even though he doesn't remember talking to her about it at all) which likely means Reina was talking about her insecurity about ayaka with her.
More likely than not they both thought the MC was cheating on Reina with Ayaka.
The mystery man could have been Natsuki dressed as a guy, or possibly her brother. MC said they were right outside his house so, unless she's stupid beyond belief, Reina absolutely intended for MC to see them together.
Either way, communication (or the absolute lack therof) pushes the story forward.