The electrode redox reactions of the Daniell cell:
$$\ce{Zn(s) <=> Zn^2++ 2 e- }$$$$\ce{Cu(s) <=> Cu^2++ 2 e- }$$
maintain at the respective electrode a particular potential where both opposite reactions have the same rate, implying there are no external galvanic causes that affect this potential. Rates of oxidation electrode reactions exponentially grow with growing potential and vice versa for reduction.This applies on potentials near the equilibrium one, when the reaction rate is not limited by the diffusion yet. Diffusion cuts off this exponential growth.The faster of the opposite reactions either pushes electrons to an electrode either pulls electrons from it until the electrode potential reaches the equilibrium potential. At this potential, rates of both reactions are equal. Even before electrodes are ever connected, these electrode potentials are in place. Electrodes do not need any external galvanic circuit.
A current flowing between 2 galvanically connected places of different potentials is not part of electrochemistry, but a part of physics which you are already familiar with.The connected external circuit is then just following the Ohm law $I=\frac UR$The current is shifting electrode potentials by moving electrons, what is the same process as in electrostatics, where potentials of capacitor electrodes are changing, when galvanically connected.
As the resulting potentials are not the equilibrium ones any more, one of the electrode opposite redox reactions takes the upper hand, leading to oxidation at the anode and reduction at cathode. Particularly, the potential of the $\ce{Zn}$ electrode with the lower equilibrium potential gets higher and starts oxidation, making the electrode the anode.
$$\ce{Zn(s) <=>> Zn^2++ 2 e- }$$
Similarly, the potential of the $\ce{Cu}$ electrode with the higher equilibrium potential gets lower and starts reduction, making the electrode the cathode.
$$\ce{Cu(s) <<=> Cu^2++ 2 e- }$$
This leads to a kind of race on the zinc electrode in providing electrons on the chemistry side and taking them by the circuitry side. And vice versa for the copper electrode. The actual electrode potential under the load is a kind of the score of this race.
Electronic chips are known for the internal use of electronic charge pumps to get different voltage levels than externally provided. E.g. once famous 8bit CPU Z80 uses external voltage +5V and 0V, internally generating also +12V,-12V,-5V. Electrodes in electrochemical cells can be considered in some sense as chemically powered charge pumps with different target potentials.
Great basic knowledge of chemistry ( and mainly of physics ) is Hyperphysics-Chemistry-electrochemistry, in form of linked cheat sheets. I would recommend electrochemistry chapters of textbooks of a physical chemistry. I have once studied a translation of Moore, EN natives would suggest much more. More advanced sources are here: SE Chemistry resources-for-learning-chemistry