Fair point, though there are ways to change the probabilities of fusion paths, just not ever fully to 0.
Reaction probabilities scale with reactant concentration and temperature in ways we can exploit.
I tried to find some numbers on the relative probabilities and fusion chains, and ran into The helium bubble: Prospects for 3He-fuelled nuclear fusion (2021) which I hope is a credible source.

This paper contains a figure, which gives numbers to the fusion preferences you mentioned.

Paraphrasing the paper in chapter "Technical feasibility of D-3He fusion" here, first we see that up to 2 billion K, the discrepancy between ²H-³He and ²H-²H fusion grows, up to about 10x. ²H-²H reactions will either produce a ¹n (neutron) and a ³He, or produce an ¹H and an ³H, with the ³H then (effectively) immediately undergoing the much more reactive ²H-³H producing a neutron too.
In addition to picking an ideal temperature (2GK), we can also further, for the price of less than a factor 2 increase of pressure, use a 10:90 mixture of ²H:³He, or even more. This will proportionally make the ²H-²H branch a factor 10/90 ≈ 11% as likely as the ²H-³He correcting for reaction crossection.
Past that, reactivity goes about with the square of pressure and the inverse of ²H concentration, so another 10x in fusion plasma pressure would net another 100x decrease in neutron emission at equal energy output.
Given how quickly fusion reactivity rises with better fusion devices, we can probably expect to work with much higher concentrations than 10:90 when the technology matures, but 10:90 at 2GK would still have about 1/100ᵗʰ the neutrons per reaction and less than 1/100ᵗʰ per energy produced compared to fully neutronic fusion like ²H-³H.

The problem is solvable, but there is definitely a potential for taking shortcuts and performing ²H-³He with much higher neutron emissions.