You mention timging. Autofocus speed and accuracy, shutter lag of DSLR and phone camera are not even comparable.
In short: moving from phone camera to an entry level DSLR with kit lenses will already be a huge step up in quality of your photos. If you understand what you are doing buying expensive lenses will add a noticeable increase to that as well. Then if you understand what are you doing and do it a lot buing a professional body will get you durability and the maximum control over the process (and ability to control aperture, shutter speed and ISO does matter too).
Yes, if you "get it" you can take a good photos with the phone if you are if conditions are favourable. But if you get it, then using DSLR will let you take more good photos, because the limits of favourable conditions are expanded a lot.
There is some truth in your last sentence, however from technical point of view, even is someone "does not get it" and would take crappy pictures on the phone then with an expensive kit he can have pictures of the same crap but with much higher quality.
That being said: The big jump is from cell-phone to DSLR (even an inexpensive one). The jump from $400 DSLR to $4000 is less noticeable.
I'm definitely not arguing that you need a high end DSLR, but even a relatively inexpensive Canon or Nikon will run circles around a camera phone.