thermodynamics - Statistical Mechanics: Boltzmann partition function

I’m a high school student and was recently studying basic statistical mechanics (for use in physical chemistry). In the derivation of the Boltzmann derivation or the partition function we arrive at a factor $e^{-\beta T}$. So I searched for the connection of $\beta$ to temperature and got one proof for that on the wikipedia site. But it seems to make to make the assumptions that

1. The energy of a system in equilibrium depends only on the temperature.


2. At equilibrium the number of microstates is maximized.


3. 
   

   It uses the formula $S=k\ln(W)$.

   
   

   So I just wanted to know that which of the assumptions is an experimental fact and whether we can arrive at the equation $S=k\ln(W)$ without already using $\beta=1/(k_BT)$ otherwise the proof would become circular.

   
   
   

To improvise, we can use the property of Lagrange multipliers to obain that $\beta=\frac{d(ln(W))}{dT}$ but I still don’t see how can we reach to the value of $\beta$ from here.

P.S. I am just a beginner so please use simple language if possible.