If they were actually interested in signing him, they could have gotten closer to each other's numbers with an honest conversation.
The approach they took, meant they were playing hardball with a lower number, which meant Mikko's agent needed to play hardball with a higher number.
This may have resulted in the Avs thinking Mikko wouldn't sign for less than $13M+. Which sounds like it wasn't true, Mikko said he didn't need more than Nate at $12.6M.
But I'm not really convinced the Avs honestly wanted to re-sign him. I think they might have already decided they'd use his cap hit to spread around the lineup. Which arguably makes sense, especially given Cale's pending raise.
In which case, an honest conversation would have been about giving a core player and fan favorite, who had been with your team though thick and thin for 10 years, and helped you win a Cup, the respect he deserves IMO.