There is one thing I have found to be "broken" with email. Or rather, a manner in which it is commonly poorly/mis-used. Group conversations.
In the work environment, little is more frustrating than not being addressed/cc-ed on some important/critical conversation. Someone leaves your address off, and "Bam!", you're out of the conversation.
I call this the "submarining conversation". It dives below the water, and you remain unaware of it unless or until it surfaces. (Whereupon it often causes considerable disruption. E.g. An entire section of specifications that you were never made aware of.)
Or you are added to an assignment and have to spend time and effort chasing down the various email conversations that have comprised its efforts to this point. AND... once you have them, gluing them together in your head and/or on screen or in text, in order to come up with a coherent and not overly redundant picture.
Google's Wave I saw as a genuine effort to address this. Actually, it is really an iteration on the centrally stored, group conversation. For example, web bulletin boards that were "all the rage" about a decade ago.
Wave added decentralize-able membership controls and sophisticated authoring and version control. Things very useful in a work or similar environment.
It didn't need to be email. But it could better fulfill a role too often filled poorly by email.
HOWEVER, Google slapped a ridiculously bogged down and unintuitive user interface on top of it, then said "figure it out". And so, it died, at least as a Google product and with respect to gaining Google's potential momentum.
SO, I'd say there are some applications of email, that could better be done other ways than through email. But this is not email's fault. It is the result of the limitations of users, both conceptually and in terms of the limited scope of tools that -- particularly in many work environments -- they are supplied and permitted.
P.S. Such organization of conversations pertains not only to contemporary work, but also to compliance and audit-ability. When trying to determine later what happened, with email conversations one it left to obtain, search, and/or reconstruct any number of email archives, hoping that none of the archives nor individual messages have been irretrievably lost or deleted.