There are other agencies that do a similar thing. Texas County and District Retirement System for example.
TCDRS.org I don't believe that one is a substitute for social security withholding however.
Essentially it's a retirement plan and any employee of any county in Texas or other taxing district can join. So these retirement systems are just pools of peoples group savings that are administered by a fiduciary agent that estimates retirement benefits based on the amount of money individuals put in using actuarial tables. They are always going to be solvent because they only pay out what an individual put in, plus a little interest. Most of these plans have a match by the employer, so to the employee the benefit is greater than what they put in.
This is a supplemental plan, but the Matagorda, Brazoria and Galveston plan is similar except for the fact that they opted out of SSA.
Is this better than social security? Not for the bigger pool of the picture. This type of opt out is very similar to the insurance industry cherry picking "healthy" clients. If the whole point is to make sure the older retirement population has some level of basic living retirement check, then a hundred million different plans spread all over the U.S. would be a disaster. But the financial sector would of course love it, since they would get admin fees on 100 million different plans.
Here is an old story on the success and failure of this plan from the NY Times dated 3/19/2005:
In Texas, A Model For Bush Proposal ; County Opted Out Of Social Security(snip)
It is an idea long promoted by free-market thinkers and others who would like to shift the responsibility and spiraling cost of retirement away from government and onto individuals, who in turn would have a shot at greater financial rewards. Critics say such an approach weakens Social Security's safety net and opens the possibility that the nation's retirement system could produce financial winners and losers. The experience of workers in Galveston offers evidence for both views.
(snip)
But the advantages are less clear for lower-paid workers. Outside experts, including researchers for the Government Accountability Office and the Social Security Administration, have found that workers earning less than $17,100 a year in 1999 would have done better under Social Security, mainly because of annual cost-of-living adjustments. The Galveston plan offers no such increases.