Why is Voevodsky's motivic homotopy theory 'the right' approach?
(Don't be afraid about the word "$\infty$-category" here: they're just a convenient framework to do homotopy theory in).
I'm going to try with a very naive answer, although I'm not sure I understand your question exactly.
The (un)stable motivic ($\infty$-)category has a universal property. To be precise the following statements are true
Theorem: Every functor $E:\mathrm{Sm}_S\to C$ to a(n $\infty$-)category $C$ that
- is $\mathbb{A}^1$-invariant (i.e. for which $E(X\times \mathbb{A}^1)\to E(X)$ is an equivalence)
- satisfies "Mayer-Vietoris for the Nisnevich topology" (i.e. sends elementary Nisnevich squares to pushout squares)
factors uniquely through the unstable motivic ($\infty$-)category.
(see Dugger Universal Homotopy Theories, section 8)
Theorem: Every symmetric monoidal functor $E:\mathrm{Sm}_S\to C$ to a pointed presentable symmetric monoidal ($\infty$-)category that
- is $\mathbb{A}^1$-invariant
- satisfies "Mayer-Vietoris for the Nisnevich topology"
- sends the "Tate motive" (i.e. the summand of $E(\mathbb{P}^1)$ obtained by splitting off the summand corresponding to $E(\mathrm{Spec}S)\to E(\mathbb{P}^1)$) to an invertible object
factors uniquely through the stable motivic ($\infty$-)category.
(see Robalo K-Theory and the bridge to noncommutative motives, corollary 2.39)
These two theorems are saying that any two functors that "behave like a homology theory on smooth $S$-schemes" will factor uniquely through the (un)stable motivic ($\infty$-)category. Examples are $l$-adic étale cohomology, algebraic K-theory (if $S$ is regular Noetherian), motivic cohomology (as given by Bloch's higher Chow groups)... Conversely, the canonical functor from $\mathrm{Sm}_S$ to the (un)stable motivic ($\infty$-)category has these properties. So the (un)stable motivic homotopy theory is in this precise sense the home of the universal homology theory. In particular all the properties we can prove for $\mathcal{H}(S)$ or $SH(S)$ reflect on every homology theory satisfying the above properties (purity being the obvious example).
Let me say a couple of words about the two aspects that "worry you"
$\mathbb{A}^1$-invariance needs to be imposed. That's not surprising: we do need to do that also for topological spaces, when we quotient the maps by homotopy equivalence (or, more precisely, we need to replace the set of maps by a space of maps, where paths are given by homotopies: this more complicated procedure is also responsible for the usage of simplicial presheaves rather than just ordinary presheaves)
Having more kinds of spheres is actually quite common in homotopy theory. A good test case for this is $C_2$-equivariant homotopy theory. See for example this answer of mine for a more detailed exploration of the analogy.
Surprisingly, possibly the most problematic of the three defining properties of $SH(S)$ is the $\mathbb{A}^1$-invariance. In fact there are several "homology theories" we'd like to study that do not satisfy it (e.g. crystalline cohomology). I know some people are trying to find a replacement for $SH(S)$ where these theories might live. As far as I know there are some definitions of such replacements but I don't think they have been shown to have properties comparable to the very interesting structure you can find on $SH(S)$, so I don't know whether this will bear fruit or not in the future.
I just want to point out, with regards to your second question, that the fact that the $\mathbb P^1$ is not equivalent to a simplicial complex homotopic to the sphere built out of affine spaces (or whichever model for an un-Tate-twisted sphere) is exactly what you would expect from Grothendieck et. al.'s theory of motives.
In fact having a homology or cohomology theory without a notion of Tate twist would be very strong evidence that we are on the wrong track!
More carefully one might point out that we want a map to etale cohomology and a notion of Chern class, which requires the Tate twisting because Tate twists show up in the Chern class map in etale cohomology.