Why does spontaneity of a reaction depend on temperature?
The issue here is in your definition of entropy. According to your definition:
$$\mathrm dS = \frac{\text{đ}q}{T}.\tag{1}$$
However, there is a small difference.
As you may know, the entropy of a system is a state function and so it only depends on the initial and final states. Therefore, if the entropy of the system were to be calculated for a specific path for which the parameters are simple to find, then the value would not change irrespective of the path.
The actual formula of change in entropy is:
$$\mathrm dS = \frac{\text{đ}q_\mathrm{rev}}{T}.\tag{2}$$
This value is calculated for a reversible isothermal process.
Now, let us assume that we have taken such an isothermal reversible process and discovered the value of $\Delta S$ and now used an isobaric process to discover the value of $\Delta H.$ Now, we solve for $\Delta G$ in our isothermal process (this is done so that we can assume constant temperature in our formula).
Therefore, for such a process
$$q_\mathrm{rev} = nRT\ln\frac{V_\ce{B}}{V_\ce{A}}.\tag{3}$$
Thus, we get change in entropy as
$$\Delta S = nR\ln\frac{V_\ce{B}}{V_\ce{A}}.\tag{4}$$
Now, enthalpy would be a constant value that we discerned from the isobaric process.
Therefore, since $\Delta G = \Delta H - T\Delta S,$ we get
$$\Delta G = \Delta H - nRT\ln\frac{V_\ce{B}}{V_\ce{A}}.\tag{5}$$
As you can see, there is still a temperature dependence.
This is because $\Delta S$  is temperature independent and so you would still have a temperature dependence for $\Delta G.$
Another thing that you may have forgotten to take into consideration is the fact that $q$ is dependent on the temperature.
The above example is that for an ideal gas expansion.
As Poutnik stated in the comments in case of a reaction, the calculation of $\Delta H$ becomes more complicated as we have to take into consideration the different molar capacities of the reagents and products which would $\Delta H$ would also be a function of temperature.
For standard value of entropy, we use the formula
$$\Delta S^\circ_\mathrm{rxn} = \sum nS^\circ_\mathrm{products} - \sum nS^\circ_\mathrm{reactants}.\tag{6}$$