After I manually added a new button, I circled back to the SDK AddIn Component Wizard Issue and figured out how to resolve.
Before posting this question, I did attempt to change the targets in the csproj file (they were incorrect) but that did not correct the issue. But that ended up being the issue.
After updating the targets in the csproj and reloading the solution in Visual Studio, I still received the error.
Then I closed the solution, and then opened the solution. I still received the error.
I had to completely exit Visual Studio 2012 and restart. And then open the AddIn solution and then attempt to add a new ArcGIS component. At this point it was successful.
My csproj change was:
<Import Project="$(MSBuildExtensionsPath)\ESRI\ESRI.ArcGIS.AddIns.targets" Condition="Exists('$(MSBuildExtensionsPath)\ESRI\ESRI.ArcGIS.AddIns.targets')" />
to
<Import Project="$(MSBuildExtensionsPath)\ESRI\ESRI.ArcGIS.AddIns.11.targets" Condition="Exists('$(MSBuildExtensionsPath)\ESRI\ESRI.ArcGIS.AddIns.11.targets')" />
and change
<Warning Text="Unable to create .esriAddin; missing ESRI ArcGIS Add-in SDK component(s)." Condition="!Exists('$(MSBuildExtensionsPath)\ESRI\ESRI.ArcGIS.AddIns.targets')" />
to
<Warning Text="Unable to create .esriAddin; missing ESRI ArcGIS Add-in SDK component(s)." Condition="!Exists('$(MSBuildExtensionsPath)\ESRI\ESRI.ArcGIS.AddIns.11.targets')" />