I think the integration with Steam should be made optional if it's kept at all.
There are numerous reasons why this can cause problems for people.
I'll go through a few off the top of my head right now, but please feel free to add if you think I missed anything. Individual problems will only be known to those having them, etc.
Steam family sharing. If you leave the client open, just to talk, or you wanted to watch a replay, or play a game - you block anyone from using family sharing until you stop. That's an annoying negative.
Steam downloads. This one definitely affects me, and i'm sure it does many others as well. I have "download while play" on steam OFF, because I simply don't have good enough internet to have that on and also play games online - I don't want to deal with annoying lag, especially because for various reasons I am generally the host of such games. This basically ensures I have to leave the client off to make sure things download. Not a great user experience.
The steam overlay has caused issues for several players i've spoken to and have mentioned it on the forums.
Preventing people from launching the steam version of the game if they have the client open.
Politeness sometimes makes people not ask people "ingame" to do anything. I can see a lot of that happening.
People who are younger and are policed by family about steam game time. Might not happen to you, but let's think about others. I'm just imagining some poor kid with 100 hours of steam time having not actually played much at all.
Beyond all of this, steam's "ingame" feature simply doesn't make sense for people in the client to have. I'm not playing a game. If someone is using the play time to help them monitor something, the amount of time they've got the client open isn't important and is actively unhelpful.
A counter in the client itself for actual ingame time would be more useful.