Calculating change in state function following different paths
You wrote, in one of your comments, &quot;My biggest question is how to demonstrate the proper use of $\Delta H = n C_p \Delta T$irrespective of the path...&quot;.
So:
In a closed system (no flow of matter across the boundaries) of constant composition (no chemical reactions),
$$dH = \left(\frac{\partial H}{\partial T}\right)_ p dT +  \left(\frac{\partial H}{\partial p}\right)_ T dp$$
$$= C_ p dT +  \left(\frac{\partial H}{\partial p}\right)_ T dp$$
$$= C_ p dT +  \left(V - \alpha T V \right) dp$$
$$= C_ p dT +  \left(V - T \left(\frac{\partial V}{\partial T}\right)_ p  \right) dp$$
Thus, at constant $p$,  $dp =0$, and
$$dH_p = C_ p dT \textit{, for all substances.*}$$
It is worth noting, as Chet Miller mentioned in the comments, that $C_p \equiv C_p(T,p)$ (i.e., $C_p \text{ is a function of both } T \text{and } p$ for real substances).
If there are no inter-particle interactions (e.g., an ideal gas), then $V = T \left(\frac{\partial V}{\partial T}\right)_ p $, and thus
$$\left(\frac{\partial H}{\partial p}\right)_ T = 0$$
You can confirm this yourself for an ideal gas by substituting $V = \frac{n R T}{p}$ for the second occurrence of $V$ in: $\left(V - T \left(\frac{\partial V}{\partial T}\right)_ p  \right)$.
Hence, for an ideal gas,
$$dH = C_ p dT \text{, always,} \textit{ even if the pressure isn't constant.}$$
In summary, if the substance is an ideal gas, $dH = C_ p dT$ regardless of whether there is a change in pressure.  If the substance is real, $dH = C_p dT$ only if the pressure is constant.

*Note that this relationship holds for all types of work.  If we add the constraint that the only work is $pV\text{-work}$, then we have the additional nice relationship that:
$$dH_p = C_ p dT = q_p \textit{, for all substances.}$$
If we keep the constant-$p$, closed system, and $pV\text{-work}$-only constraints, but remove the constant-composition constraint (allowing chemical reactions), then it will no longer be the case that $dH_p = C_ p dT$, but it will be the case that:
$$dH_p = q_p \textit{, for all substances.}$$