I don't know about gamma function, but if I were to extend the definition of factorials in an intuitive and natural way, I would do it like this:
Suppose we want to get the value of $5.5!$.
So, I need to get in the middle of $5!$ and $6!$ intuitively.
To get to $6!$ from $5!$, we multiply $5!$ by 6, i.e. we apply the function $f(x)=6x$ to $x=5!$.
Since, we have to get in the middle of this operation, I'd apply the funcional-square root of $f(x)=6x$, i.e. $\sqrt{6}x$ to $5!$ so, $5.5!=5!*\sqrt{6}$ by this definition.
Similarly, To get $7.1!$ I would apply the functional-tenth root of $f(x)=8x$, i.e., $f(x)=x*8^{0.1}$ to 7! which gives $7!*8^{0.1}$.
So, my extension would be: To get $x!$: If $k$ is the fractional-part of $x$ and $a$ is its integer part, then $$x!=a!\cdot (a+1)^k$$
I couldn't understand much about gamma function, but I can surely say one thing that this one is a lot simpler. Is there anything wrong with this?
UPDATE: I did some calculations on my calculator and found that my factorial definition gives values relatively close to the gamma function.
See https://proofwiki.org/wiki/Gamma_Function_is_Unique_Extension_of_Factorial
Essentially, once you want alternative extension of factorial, you lose either factorial-like behaviour, or lose log-convexity, and both are very commonly used property of factorial (or you lose analytic, which is probably even worse).
Also from number theory point of view, Gamma function is much more useful due to L-function.