To understand Thomas's logic, start with Section I of the dissent: https://www.supremecourt.gov/opinions/16pdf/15-1256_5i36.pdf. Thomas is not saying that the state should be able to "keep money from the innocent." Thomas's reasoning is roughly the following:
1) The 14th amendment requires the government to give you adequate process before depriving you of a property right.
2) At the time petitioner was convicted, he paid over a sum of money to the state. He lost the property right in that money, and the state gained the property right in that money.
3) After the conviction was overturned on appeal, the state required petitioner to go through a process to get his money back. Petitioner alleges that the process was inadequate under the 14th amendment.
4) To make a 14th amendment deprivation claim, you have to show deprivation of a property right. At the time petitioner sought his money back, he had no property right. That transferred to the state under step (2).
5) Thomas asks: where is the property right that is the basis for the 14th amendment claim? It can't come from the 14th amendment itself, because that only comes into play once petitioner has a property right.
To get around (5), you can theorize that the initial payment automagically become null-and-void when the conviction was vacated. But that would be an unusual result, because legal judgments don't ordinarily have that kind of effect. Say you buy some property, then sue the seller and get a judgment saying you overpaid. You can assert that judgment against the seller to collect, but at most the judgment means the seller owes you money back, it does not automatically transfer the property right in some of the money you paid back to you.
I think the majority is right in that the overall effect of the Colorado law would be a due process violation. But the majority's reasoning kind of requires thinking of the whole process as a black box and not thinking too hard about what happens inside.