No, he is quite correct. The FMS gives you net climb, as you say, which is what you are required to show will clear all the obstacles - and which gives you a small margin on what would otherwise be a very close pass of them if you actually flew the net path!
In theory if you know where all the obstacles are you can feed them all into the FMS (one by one) and check that you will clear them. The requirement to use a performance software stems solely from the fact that without the database in there it is impossible ot know where all the obstacles are, and which route to fly to avoid ones you can't necessarily clear if you climb straight.