I am not really familiar with homotopical category theory, so please forgive me if I make rude mistakes. I know quite a bit of common category theory, as well as familiar with algebraic topology.

How can one construct exact sequences abstractly in the framework of homotopical categories? Consider any model you like (topologically or simplicially enriched, or $(\infty,1)$-categories). nLab teaches us that to construct exact sequences in topology one needs to consider fiber sequences and homotopy limits/colimits. A fiber sequence $F\to E \to B$ is a homotopy pullback

\begin{array}{ccc} F & \rightarrow & E\\\ \downarrow & & \downarrow\\\ \ast & \rightarrow & B\\\ \end{array}

The loop space $\Omega X$ of $X$ is defined as a homotopy pullback

\begin{array}{ccc} \Omega X & \rightarrow & \ast \\\ \downarrow & & \downarrow\\\ \ast & \rightarrow & X\\\ \end{array}

Dually, one can define cofiber sequences and suspensions $\Sigma X$. We get a standard adjunction $\Sigma \vdash \Omega$. Given a fiber sequence, one can have a long fiber sequence $\dots\to\Omega E \to \Omega B \to F \to E \to B$. To begin with, consider a homotopy pullback rectangle

\begin{array}{ccc} \Omega B & \rightarrow & \ast \\\ \downarrow & & \downarrow\\\ F & \rightarrow & E\\\ \downarrow & & \downarrow\\\ \ast & \rightarrow & B\\\ \end{array}

Standard diagram chasing and (homotopic) universality arguments show that we get a fiber sequence $\Omega \to F \to E$ etc. Now if we apply the functor $\pi_0 \simeq \pi(\ast,-)$, we get a sequence $$\dots \to \pi_0(\Omega E) \to \pi_0(\Omega B) \to \pi_0(F) \to \dots$$

Using the fact that $\pi_0$ preserves fiber sequences (because it is representable) and $\pi_0(\Omega^i X) \simeq \pi_i (X)$ we get a classical long exact sequence of homotopy groups for a fibration.

Now things get trickier if we consider homotopy pushouts instead of pullbacks. If we have a cover $(U,V)$ of space $X$, then we have a homotopy pushout diagram

\begin{array}{ccc} U\cap V & \rightarrow & U \\\ \downarrow & & \downarrow\\\ V & \rightarrow & X\\\ \end{array}

Standard arguments allow to replace this diagram with cofiber sequence $U \vee V \to X \to \Sigma (U\cap V)$. In theory it should give Seifert-van Kampen theorem and Mayer-Vietoris sequences in homology and cohomology in a similar way. In practice, I don't understand how can this happen. To get Seifert-van Kampen theorem, one should apply $\pi_1\simeq \pi_0 \circ \Omega$, but $\Omega$ fails to preserve cofiber sequences. Dold-Thom theorem tells us that $H_i = \pi_i \circ S$, where $S$ is the topological abelianization functor. We're fine with $S$, because it's a left adjoint to forgetful functor, but we have the same problem with $\pi_i$. It doesn't preserve cofiber sequences. Even if it did, we would have terms like $\pi_i (S \Sigma X)=H_i (\Sigma X) = H_{i-1}(X)$ for $i>0$, and that is nothing like an ascending sequence of homology groups. Even cohomology isn't fine: we can take the space of maps to $K(\mathbb{Z},1)$ and it turns cofiber sequences into fiber sequences, but we still have a problem with grading, like in case of homology.

So my question is: how can we deal with these difficulties and get the mentioned theorems? I didn't have any success till now in finding references on homotopy limits/colimits technique, although many sources mention them and deal with technical properties, like their existence in model categories. Any references would be greatly appreciated!