Even if you were to amend Article 16 to permit a majority to pass a budget, you'd still be faced with the impassable wall that Prop 13 represents.
Section 3. From and after the effective date of this article, any changes in state taxes enacted for the purpose of increasing revenues collected pursuant thereto whether by increased rates or changes in methods of computation must be imposed by an Act passed by not less than two-thirds of all members elected to each of the two houses of the Legislature, except that no new ad valorem taxes on real property, or sales or transaction taxes on the sales of real property may be imposed.
So a majority could PASS a budget, but they would be handcuffed in that they could only do so within the bounds of CURRENT state revenue. In our current budget situation, that would offer little help. Our party would have the ability to better define where the cuts would happen, but we would still be facing slash and burn budgets with no new revenue. In order to raise taxes or fees to actually PAY for that new budget, you would still need a 2/3'rds majority per Prop 13. You can't raise sales taxes, income taxes, or property taxes under Prop 13 without a 2/3rds majority.
As for support, I have to remind you that Prop 56 tried to repeal these requirements in 2004 and only garnered about a third of the vote...it wasn't even close. Californian's, even liberal ones, have long suffered a cognitive dissonance when it comes to state taxes and services. We want those services, but we don't want to pay for them ourselves. Polls have repeatedly shown that there is a statistical dead heat in California on the 2/3rds requirement for budgets, but that the 2/3'rds requirement for taxation increases still has fairly solid support among voters. Even the Republican leaders have publicly said that they'd give up the 2/3'rds budget requirement if they had to, but that they'd fight like hell to keep the 2/3'rds tax rules. They know that's where the real power, and problem, with our budget is located.
There's no advantage in passing a budget you can't pay for.