Why do you use discount rate X in the valuation models?

The valuation models are designed as something that users can tweak based on their assumptions (e.g. different discount rates) as clearly it's impossible to have a "one size fits all" pre-baked valuation model across every stock in the market. In terms of the discount rate assumptions, the rates are just indicative based on a general view that small-caps tend to be more volatile and riskier, and clearly there are exception to that notion (e.g. the banks in recent years).

Another option would be to calculate the Cost of Equity as the Risk free rate of return + Beta x (market rate of return- risk free rate of return), but we are very sceptical about the value-added of Beta despite its widespread usage in the City.

The reason for this are clearly set out in James Montier's paper "CAPM is CRAP (completely redundant asset pricing)" and another one on the Efficient Market Hypothesis. It's a chapter in his excellent Value Investing book which we'd recommend.