But there is always some metric to be gamed. There is always some causality chain which results in a specific distribution of money to organizations. Just because we close our eyes, it does not make the causality go away.

This doesn't necessarily mean that the system is gameable. If we suppose that what we're measuring is actually exactly what we want to get out of the program, then the only way the program can get ahead is by providing more of what we want. The system can only be "gamed" to the extent that there's a divorce between what we want from it, and what sort of output we measure.

I'm leery of organizations providing their own statistics on how effective they are, which may just be another form of lobbying and propaganda. I'd lean towards carving out from the budget a group that independently assesses effectiveness of each of the organizations. It's admittedly imperfect, but it would be more impartial than what seems to be in place now, and agencies wouldn't necessarily be at a disadvantage for lacking their own internal measurement tools. That still leaves the problem of choosing the right metrics. Something simple like budget percentages and ratios would be a good place to start. There are a lot of hard-to-compare types of services out there; after school programs aren't like Meals on Wheels programs. It's hard to come up outcome based metrics to say which service is better than another when there's so many different categories. Adopting something along the lines of the financial ratings at Charity Navigator could at least get everyone on the same page for controlling costs at their organizations.

Instead of organizations wasting money on and time a dysfunctional official metric, there will be organizations wasting money and time on alternative ways (bribery, fraud, advertising, lobbying, propaganda...) of convincing government that it's they who should get a bite from the budget.

Sure, but the existence of bribery and corruption can actually be viewed as an argument for not establishing any rigorous metrics: the worse things are, the more likely it is that any supposedly objective metrics get turned into instruments for giving specific organizations plenty of money and shutting everyone else out. That happens all the time in the public sector, with e.g. lucrative contracts being supposedly offered for anyone who can fill the requirements and bids the lowest, but with the requirements being intentionally crafted so that only a few favored organizations can match them. Without such a setup, other organizations might actually get their offers into consideration.