I think the difference you are picking up on has to do with the structure of the equation itself. In some equations, the unknown $x$ is acted on serially through a sequence of nested operations; solving the equation amounts to "unwinding" those operations one by one. For example, $\frac{(3x+5)^2-19}{2}=10$ can be thought of as "take $x$, triple it, add $5$, square the result, subtract $19$, and divide by $2$; the result is $10$. You can diagram this as a series of functions like so:
So to solve it, you reverse those steps: Start with $10$, multiply it by $2$, add $19$, take the square root, subtract $5$, divide by $3$.
But in other equations -- even fairly simple ones -- have a different kind of structure. For example, $x^2+5x=10$ looks like this:
Notice that this structure resists any attempt to solve it by "unwinding", precisely because of the fork in the diagram. The $x$ flows through more than one path, which makes it impossible to trace the result backwards to its source.
Some equations are presented in a form that at first appears to have multiple-paths (e.g. $3x + 5x = 16$) but we can rearrange them to a single-path structure (for example, via the distributive property / combining like terms). But higher-degree polynomial equations typically have terms that cannot be combined, and this is what makes them resistant to the kind of intuitive solution you are asking about.
Update: In the comments below, the OP asks:
This makes sense, but then how are general solutions for higher order equations like the quadratic formula derived algebraically/symbolically? I am familiar with the geometric proof by completing the square, but do higher order equations suddenly require the vantage of geometry to solve? Or is there a pure/direct algebraic derivation that can be thought to extend from the basis of "unwinding" or the like?
Although completing the square is historically geometric in origin, it can be understood in a purely algebraic way as a method of restructuring a function so that it has a "serial" structure, enabling it to be solved via unwinding. Let's take the example of $x^2+5x=10$, already diagrammed above. In completing the square, you first add $(\frac{5}{2})^2=\frac{25}{4}$ to both sides of the equation, obtaining $x^2 + 5x + \frac{25}{4} = \frac{65}{4}$. Then you recognize the left-hand side as a perfect square trinomial, so the equation can be written $(x + \frac{5}{2})^2 = \frac{65}{4}$. This equation, if represented diagrammatically, would have a simple serial structure: Start with $x$, add $\frac{5}{2}$, square it, and end up with $\frac{65}{4}$. It can then be solved by unwinding: Start with $\frac{65}{4}$, take the square root(s) (keeping in mind that there are two square roots, one positive and one negative), and subtract $\frac{5}{2}$.
Of course, this is not the only method that can be used to tackle quadratics. Consider the slightly different example of $x^2 + 5x = 24$. If we rearrange this as $x^2 + 5x - 24 = 0$ and factor the LHS, we get $(x-3)(x+8)=0$. This equation can be diagrammed as follows:
At first glance this looks to be no better than the original equation; it has a fork in it, and seems resistant to unwinding. But! There is this property of real numbers, the "zero product property", which says that if two numbers multiply to be zero, then one of them must be zero. And that allows you to split the diagram into two separate diagrammatic cases:
And each of those can be tackled via a very simple unwinding method.
In short, most of the techniques that are taught (at least at the high school level) for solving polynomial equations can be understood as "methods for re-structuring equations so that unwinding techniques can be employed". (I'm not claiming that they are taught in those terms, or that they should be, but merely that they can be thought of that way.)
Unfortunately this only gets you so far. Once you get to 5th degree polynomials, it is a famous result that there may be solutions that cannot be expressed by a combination of "simple" operations (see here). That means, among other things, that there is no way to restructure a general 5th degree polynomial so as to enable a solution via unwinding techniques.
Best Answer
If the definition of $y = \sqrt x$ is "that number $y$ for which $y^2 = x$", and if you also agree that (positive) square roots are unique (there is only one; any two positive numbers $y_1$, $y_2$ with the property that $y_1^2 = x = y_2^2$ must in fact be the same, $y_1 = y_2$), then it is not hard to prove that $\sqrt a \sqrt b = \sqrt {ab}$.
Here is a proof: if you claim that $y_1 = \sqrt a \sqrt b$ and $y_2 = \sqrt{ab}$ are in fact the same, you can demonstrate it by showing that they satisfy the same property; namely, $y_1^2 = y_2^2 = ab$.
Indeed, $y_1^2 = (\sqrt a \sqrt b)^2 = \sqrt a^2 \sqrt b^2$ (by commutativity of multiplication), and $\sqrt a^2 \sqrt b^2 = ab$ by definition of square root.
Likewise, $y_2^2 = \sqrt{ab}^2 = ab$ by definition of square root.
Hence $y_1^2 = y_2^2$, ie each of $y_1$ and $y_2$ satisfy the property of "is a square root of $ab$"; since (positive) square roots are unique, then $y_1 = y_2$.
You can present an even shorter proof once you are savvy with definitions; observe $(\sqrt a \sqrt b)^2 = \sqrt a^2 \sqrt b^2 = ab$, then immediately by definition we have $\sqrt{ab} = \sqrt a \sqrt b$.