How powerful is a k-stack pushdown automaton with a unary stack alphabet?
Such automata are called counter-automata. Each counter holds a natural number, which can be tested for zero. (So your UPDA has a single stack symbol plus a way to recognize the empty stack.)

Two push-downs are Turing complete, i.e., they can compute any Turing computable function. Basically, because two stacks can simulate a queue. So you have proved yourself that 3-UPDA is Turing complete. But actually two counters are enough. The trick is to code two numbers $(i,j)$ into a single number $2^i 3^j$ and doing your own construction again.

The family of languages is called one-counter languages, which have some fame in formal language theory. I browsed my own old answers and found: Which languages are recognized by one-counter machines?

Also: The simulation of TM's by counter automata is explained in the old textbook by Hopcroft and Ullman (Intro to Automata Theory, Languages, ...) in two steps. First using four counters and coding the contents of (half of) a working tape of a TM by a single number. The result is attributed to Minsky (1961).