The average scoring system has never come close to working for many reasons. 9 holes doubled up, sandbagging opportunities, trapdoor tier advancement, etc..
It needs to be a lot more complex to be a fairer playing ground (and maybe confuse the simpler minded `bagger :) ) and advancement will not come as a shock to the innocent who plays the same easiest 9 holes without knowing the pitfalls underfoot.
Counting only tournament scores doesn`t address the "unlimited" tourney issue. Does only the best score count, thereby closing out the `bagger, but boosting the noob who has tenaciously worked for that one good win? How much would an "unlimited" score be weighted towards the average? And so on.
As for the equipment, if it is not tier regulated, then it could be "tee box" regulated. For example, if a pro with master club/balls wants to use them, then it`ll be from further back. This would only be enforced for tournament and ranked rounds (including match play). These rounds also need to be weighted separately for fairness to deter sandbagging.
Another possibility is to have different averages applied to the different modes of play. I`m already getting confused with this :). Just my two cents