Why does the LTspice op amp stability tutorial recommend open loop analysis?
Yes - sometimes the terms are somewhat misleading. In general, we have three different gain conventions for feedback circuits:

Open-loop gain of the amplifier (alone): Aol,
Closed-loop gain of the amplifier with feedback: Acl,
Loop gain Aloop, which is the gain of the complete loop (to be measured or simulated after breaking the loop at a suitable point). Hence, the loop gain is the product $$A_{loop}=(-A_{ol}*\beta)$$
with beta = feedback factor
Note that this definition includes the inverting sign at the opamp input.
Based on these conventions the closed-loop gain is $$A_{cl}=(A_{ol}*\alpha)/(1-A_{loop})$$with alpha = forward factor, if existent, otherwise unity

Note that stability margins (phase margin, gain margin) are defined for the loop gain Aloop only. For the purpose of determining the margins, the loop must be opened at a suitable point (opamp output or inv. input) for injecting a test signal. I hope this clarifies something.
It is the purpose of the following example to demonstrate the meaning (and the correct sign) of the term&quot;alpha&quot;:
Example:
(a) Non-inverting amplifier (R2=feedback resistor): alpha=1, beta=R1/(R1+R2);
(b) Inverting amplifier: alpha=-R2/(R1+R2), beta=R1/(R1+R2).