I'm just thinking about what sort of things my DS (just turned 4) responds to...

- During the tantrum are you getting into a screaming match with him, or do you ignore it?

From the outside and with the limited info you've given, it sounds to me like he wants the attention.
While you were using the rewards chart he was getting positive attention and positive reinforcement - clearly that's what he needs and respond to.

My DS is really excited when we give him responsibilities and he gets to feel like a 'big boy'.
Since having DS2 9 months ago, we have created a shelf in the pantry for him so he can get his own snacks (he stills asks when he can have one), he is then free to choose anything from the shelf. As of this week, I have bought a small cereal container and put rice bubbles in there so he can self serve breakfast. BUT the condition was, he has to wake up and brush his teeth and get dressed and then he can make his breakfast ALL BY HIMSELF! He thinks this is the greatest thing ever, and he is finally getting in super speedy mode - it used to take 20 minutes of arguing and fluffing around.

Anyway, my point is, giving him responsibilities and trusting him to do some things, gives him sooo much confidence and has really improved his behaviour. Sure, he spilt some milk this morning, but so what - he actually tried to clean it up by himself saying he could do it.

I am by no means a 'perfect' parent, nor am I always (read rarely) patient... And I yell as much as the next person, but my son is also a really good communicator, so I find that if I talk to him and try to treat him like an equal and not just a 'child', then he responds heaps better.
We negotiate and talk things through with him and he seems to appreciate that.

May or may not help... But I just think you probably need to try something new to break the tantrum routine/cycle.