They don't collapse into two-valued logics, but they can be encoded into two-valued logics in the same way that numbers can be encoded into sets.
The simplest way to see this is to think of a computer. Its operation can be described by a function in two-valued logic, taking the bits making up the state of the CPU and memory as input then returning the next state. And you can write programs that do multi-value'd logic... so within the two-value system you can make a little multi-value system.
In terms of the database analogy: you have to translate the data and the queries. So you can't query "what is their age", you can only query things like "the current time is X, what is their age".
I agree that you can model a multi-valued logic system on a universal Turing machine but you are wrong to think that doing this is the same as mapping to classical logic. TMs do not intrinsically follow the rules of classical logic, rather, they obey classical logic (when they do) because they have instructions written into their head. Similarly, the marks on the tape do not intrinsically represent two-valued logic even if there are only two kinds of marks on the tape. This has nothing to do with succinctness or making Turing Machines better and everything to do with 'what do the marks represent?' and 'what is the best way to reason with the marks?' which are definitely interesting questions in my book.
The simplest way to see this is to think of a computer. Its operation can be described by a function in two-valued logic, taking the bits making up the state of the CPU and memory as input then returning the next state. And you can write programs that do multi-value'd logic... so within the two-value system you can make a little multi-value system.
In terms of the database analogy: you have to translate the data and the queries. So you can't query "what is their age", you can only query things like "the current time is X, what is their age".