Is Cobb-Douglas the only output function corresponding to a competitive economy?

Apologies if this is a rather simple question, I appreciate any guidance.

$$Q(K,L) = AK^\alpha L^{\beta}$$

where A is a constant. Identify the conditions on $\alpha$ and $\beta$ for output function Q to correspond to a competitive economy.

So a competitive economy is one where the factors of production are paid their marginal product. I am aware that the Cobb-Douglas production function alignes with this, so I would have: $\alpha + \beta = 1$ and $\alpha, \beta > 0$.

But is this really the only and most general answer? I cannot see why we couldn't have different exponents and still pay the factors of production their marginal product.

Any constant returns to scale function is compatible witha competitive economy. Cobb-Douglas is not the only one. Google CES production function.

Also, product can be wasted or be an externality, so even in non CRS competitive economy can exist.

Let $a+b<1,\;\; a,b>0$. Pay the factors of production their marginal product:

$$rK = \frac {aQ}{K} K= aQ,\;\;\ wL=\frac {bQ}{L} L=bQ$$

So total payments to factors of production will be

$$rK +wL = aQ + bQ = (a+b)Q < Q$$

And the question is: who gets the rest of the output that has been produced?