Neither of my carbed MC's need 5-10 minutes to warm up even in cold weather. They are running smoothly after long enough for me to get situated, put on my helmet and gloves, look around, check to make sure the engine is running correctly, set the choke (enrichener) if the temperature is cold out and go. This can't be more than two minutes. After a short period of time maybe less than a mile, no matter what the temperature is outside I put the choke to off and ride. My in-line four is a little bit more cold blooded than the Savage, but most days I could easily start the Savage with no choke and go. Personally I don't ride my MC's hard, so since the first couple of minutes is neighborhood riding, I suppose it could be considered an elongated warm up.

The short answer is yes it's the carb. Your fuel injected bike allows you to take off as soon as it is started. Sensors and an integrated ignition mapping allow for this.

The only reason to delay for a minute or two, is to give the engine time to circulate the oil enough so that all parts are freshly coated before a load is applied.  Within that minute or so it really does not matter whether the carb needs a little choke or not...it is all about the oil.  A fuel injected engine will run perfectly from the moment it is started so you really would not know if it was cold or warm based upon the way it runs.  Just let the oil circulate and then go.