I checked out some resources on how the constant of proportionality of the Coulomb force was discovered and to my surprise, I found out that it was mathematically derived (unlike the Cavendish experiment for the gravitational constant) by using Gauss's law.
When I searched for the proof of Gauss's law, it used Coulomb's law WITH THE COULOMB CONSTANT AS $1/4πε$. Surely I am missing something, can you guide me?
Links - For Coulomb's constant, check out the 'Value of the constant' section https://en.wikipedia.org/wiki/Coulomb%27s_constant
For proof of Gauss's law, check out 'Deriving Gauss' law from Coulomb's law' in 'Relation to Coulomb law' section https://en.wikipedia.org/wiki/Gauss%27_law
Answer
You might want to read https://hsm.stackexchange.com/q/3553 for the history part.
As far as the confusion regarding Gauss law and Coulomb's law is concerned, you really can't prove either of them independently. And it makes sense, as Coulomb's law or Gauss law describe something related to the real world, and not something that could be logically or mathematically deduced. You can't tell whether masses attract just by logic. Similarly, you can't tell if Coulomb's/Gauss law is true by mere mathematics. You have to do experiments and draw conclusions from their results.
Then why do we even consider them true? That's because no experiment till date has given a result that defies them (that's pretty much the case with almost all physical laws which are considered true).
So, if you consider Gauss law to be more fundamental, then Coulomb's law is its consequence, and vice versa.
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