In pretty much all introductory chemistry courses, the galvanic cell (i.e., battery) is introduced as a system of two half-cells with a salt bridge converting chemical energy to electrical energy. From what I understand, this is what is called a Daniell cell. For example, $\text{Zn}_{(\text{s})}|\text{Zn}^{2+}_{(\text{aq})}||\text{Cu}^{2+}_{(\text{aq})}|\text{Cu}_{(\text{s})}|$ with an $\text{NaCl}_{(\text{aq})}$ salt bridge.
However, we can design a much simpler battery using only one cell: $\text{Zn}_{(\text{s})}$ and $\text{Cu}_{(\text{s})}$ electrodes both dipped in the same $\text{H}_2\text{SO}_{4\text{(aq)}}$ solution (similar to a lemon battery). Both designs would produce the same voltage (~$1.10 \text{V}$). But since the latter design is simpler/more practical/cheaper/etc., why do most resources exclusively talk about the Daniell cell? Are there flaws in the single-celled system?
I tried looking for an answer in Why can't a galvanic cell be a single cell? and Why can't an electrochemical cell be a single cell?. They both point towards the idea that single-celled batteries would not do useful work on the external circuit, but I'm not convinced. In the single-celled design, when $\text{Zn}_{(\text{s})}$ dissolves to $\text{Zn}^{2+}_{(\text{aq})}$, it won't react with $\text{Cu}_{(\text{s})}$. Also, free electrons can't flow in an aqueous solution. Hence the only path for electrons is through the wire. Am I missing something?