Here are my comments on the associated chemistry not usually discussed in presentations of the Daniell cell (see, for example, this overly simple rendition).
The electrochemistry starts with a divided galvanic cell with the zinc displacing the copper metal from the aqueous CuSO4 (see prior reference for half-cell reactions).
Importantly, the concentrated solution of copper sulfate acts both as an electrolyte and more, per this source, where MagLab comments:
A container lined with copper (the cathode) forms the outer portion of this device, and is filled with copper sulfate, an electrolyte that reacts chemically with the solid copper.
My supposition on this comment is the following reaction:
$\ce{Cu + Cu(II) ⇌ 2 Cu(I)}$
The above often referenced reaction, per my research, suggests that it can apparently proceed more rapidly in acidic conditions.
Note, as to your question of what happens upon inserting a copper rod into the cupric sulfate solution, above is what could occur over time with respect to standard chemistry (and, not electrochemistry, so forget charge issues).
In the current context, cuprous sulfate is likely created, which is unstable decomposing on warming and, although Cu2SO4 is water-soluble, it reputedly undergoes decomposition in aqueous conditions, per Wikipedia (which is the reverse of its formation reaction above).
Operationally in the presence of air/oxygen, a possible electrochemical reaction (well known for ferrous) between the formed cuprous and oxygen in acidic conditions (which apparently take places with even other transition metals than Fe and Cu, like Mn, Co,...) could also be occurring (albeit limited depending on surface area exposure to air) with time:
$\ce{4 Cu(I) + O2 + 2 H+ -> 4 Cu(II) + 2 OH-}$
This actually introduces a second electrochemical reaction, which may importantly regenerate the CuSO4 (as could perhaps the simply application of heat). Maintaining a high copper sulfate concentration is likely beneficial to the efficient operation of the battery cell. In practice, I would advise, adding H2SO4 to the CuSO4 to maintain an acidic pH, as H+ is consumed per the above reaction. Interesting, per Wikipedia on the original Daniell Cell, to quote:
The copper vessel was filled with sulfuric acid solution saturated with copper sulfate to above the level of the perforated disc.
Over time, copper buildup will block the pores in the earthenware barrier and cut short the battery's life.
The above-cited issue from Wikipedia on the Daniell cell with respect to blockage caused by copper is a problem, but perhaps addressable, in part, by mechanically applied scrubbing.
An interesting experiment for some may be to amend the battery cell with an efficient oxygen interface (as found in metal-air batteries) and record changes in performance.
Note: there are other examples of electrochemical and other chemistry occurring together as in, for example, the action of oxygen on copper in a solution of ammonia. See this source, as an example: "Kinetics and Mechanism of Copper Dissolution In Aqueous Ammonia", (source link).