I actually think Tsutomu's family took an important roles in here, as they're the ones that made him remember that he actually know Yuzuru from the past, which leads him to feel guilty and now he wants to redeem himself to Yuzuru. I think the next chapter will talk about Yuzuru's perspective like when did he remembered Tsutomu is his childhood friend, and when did he start to like him. I think Yuzuru's identity might be confusing, but I think the author wants to make it self-explainetory, because he said that he worked on a project about the dating sim and that Tsutomu is a client, so we can assume that Yuzuru might be working in a game company and that he probably agreed to be a model for the game character.
As for the rumors, Tsutomu probably assumes that the company president spreads those rumors because he fancies Yuzuru.
Hope that explains some of the confusion (づ￣ ³￣)づ
Sorry tho if it's not that good, english isn't my first language t-t
Many small details were not explained properly like the friend who works for a game company and how the rumors of Yuzuru dating the director spread.
I think it's partly due to bad translation, the dialog was messy sometimes, but the author didn't help either.
Chapter 5 should be a climax and enlighten us about Yuzuru's identity but instead the author introduced two new characters that were not important or at least didn't have to be shown for us to understand the story.
In chapter 6 we'll probably get the explanation of how Yuzuru became a character in the game, that could have been done in chapter 5 instead, leaving more pages to conclusion and romance. Now chapter 6 will probably be rushed.