For example, if a customer emails firstname.lastname@example.org, you get an email notification about the new conversation, but you can't keep the conversation going solely through your email client. You have to use Front.
This is a dealbreaker, I think. The normal day-to-day interactions — ie., responding to mail — that Front handles can, and should, easily be done with an email client; there is nothing different about Front does that an email client doesn't do today. People can still use the app, but they shouldn't be forced to switch.
I've found it hard to pull myself away from Gmail and use these apps though, old habits are hard to change…
This seems like a nice program that we definitely may need in our current company (we do internet-based loans and use a support@ email which is shared between several people.
Unfortunately, it is not clear what features your service can provide; what problems it can solve. I opened an account and played a bit with it but just couldn't find something that I could show my colleagues to tell them "look, we need to use this service"
I would recommend showing one of those 2 minute video marketing overviews of what we can achieve with your service. If you put that in the homepage, it will be easier for people like me to sell this to other colleagues.