A question about the equality of heat and changes in enthalpy
In the absence of non-PV work (at constant pressure), the change in enthalpy of the system is equal to the heat exchanged between system and surrounding. This follows from the first law of thermodynamics and the definition of enthalpy.
The equation you are showing is based on the definition of the Gibbs energy (the chemical potential is defined via the Gibbs energy, and the Gibbs energy and the internal energy are different by TS). Just because you define some new thermodynamic functions and express the change in enthalpy with it does not invalidate the first law of thermodynamics.

How can we just ignore the summation for something like a chemical reaction?

Here is a silly calculation to model what is going on. Let's start with a variable $a$ known to be five:
$$a = 5$$
Now we add $b$ to both sides:
$$a + b = 5 + b$$
Let's define a new variable to simplify the right hand side:
$$c = 5 + b$$
and substitute that into the equation:
$$a + b = c$$
Now we solve for a:
$$a = c - b$$
It looks like $a$ depends on $b$ and $c$. However, we know $a$ is equal to 5, so it is constant. If you substitute the definition of $c$ back into the bottom equation, you get back to $a = 5$.