Unfortunately, there doesn't exist such a formula, which exactly solves your problem. Noise, interference and susceptibility always tremendeously depend on your actual setup, on cable lengths, shielding measures and your grounding techniques, especially if radiated interference is the main issue.
It's a good design practise to make an application as immune to undeterminable parameters, like cable lengths for instance, as possible. This is done by shielding, grounding and filtering. Shielding and grounding prevent fast electrical fields (high dU/dt) from injecting charge into sensible points, by minimizing involved stray capacitances. And filtering prevents spiky currents from forming bigger current loops than necessary.
But with high current applications proper filtering might become difficult, because magnetic materials saturate and capacitors become large and unreliable. In such cases the develop of interference must be 'eliminated' directly at the source, for instance by avoiding the creation of fast edges, means by using the methode of zero-cross switching.

If it's impossible to give an exact formula because of too high complexity, then it's always useful to analyze much simpler situations and to derive useful conclusions by physical intuition.

How to convince your friend, that zero-cross switching is also useful with resistive loads?
For simplicity we don't want to analyze sine waves, but square waves of different rise times. The use of square waves has the advantage, that its spectrum is very well known, so that no Fourier analysis has to be done.
Assume that you turn-on and -off a resistive load controlled by a square wave function with a period of several mains sine's periods. Furtherly assume, that the rise time of turn-on current is either very very small, only limited by the finite turn-on time of triac or what else. This case represents the turn-on of a resistive load at the moment of maximum mains voltage (230V x 1,414 = 325V), means the total opposite of zero-cross switching. If the rise time of turn-on current is "Tr", then

dI / dt = (dU / dt) / R = 325V / (Tr x R)

where R is the resistive load.

In the other case rise time of square-wave-like turn-on current shall be identical to that of zero-cross switching. Rise time in the zero-cross moment we get by taking the first derivative of mains sine 325V x sin (w x t) at t = 0, which gives 325V x w x cos (w x 0) = 325V x w, where w = 2 x pi x 50Hz. Rise time of turn-on current then is

dI / dt = (dU / dt) / R = 325V x w / R

Now we can compare both rise times by making the division:

(325V / (Tr x R)) / (325V x w / R) = 1 / (w x Tr)

So, if you turn-on R in 1µsec, then rise time of current is about 3180 times smaller than that of zero-cross switching!

Why is it so important to achieve a high rise time, means to turn-on the current slowly?
This we will see, when having a look at spectrum of square wave: From the fundamental the harmonics fall with a slope of 20dB / decade. But finite rise time gives an additional fall of harmonics with a slope of 20dB / decade above a corner frequency of

fc = 1 / (pi x Tr)

So, between the fundamental and fc a slope of 20dB / decade occurs, and above fc we get a slope of 40dB / decade.

The reason why fc should be as low as possible for high current applications, where filtering might be too sophisticated, is, that the higher harmonics of current become transferred into electromagnetic radiation, when running along sufficient long conductors! If the conductor length is 1/6 of wavelength, then conducted electricity becomes transferred into electromagnetic radiation, means enormeous amounts of radiated interference can develop.

An example:
A 15 meter long cable becomes an antenna for frequencies above 3MHz. If rise time of turn-on current is 1µsec, then fc = 318kHz. For high current applications fc might not be below enough 318kHz, so that relevant radiated interference can develop. But if you use zero-cross switching, then fc is 3180 times lower and much much less power is pumped into electromagnetic radiation!

Again, whether zero-cross switching is needed or not, depends on the height of switched current and on whether any filtering is achievable or not. Nevertheless, all todays engineers are responsible for not contaminating our electromagnetic resources more than ever necessary!

Kai