The prejudice will be there anyway, not explained by ASD, but by just ... whatever bad label comes to their mind. It would surface in education, social life, relationships, employment.
The other consideration is that the child could have a crisis later and without a dx it would delay help even further.
Every parent is right to consider the decision, any decisions.
But I think the very question of cons of the diagnosis might be a reflection of parent's own frame of reference. I don't mean to attack anyone. I had to go through the process of adapting my thinking to DC diagnosis. I had prejudice against disability, mental disability in particular. I wasn't aware of it and of why it was wrong. Basically I never given it a thought before it happened to me. I went through a steep learning curve to understand how DC are different, what this means. It means that 'normality' is more arbitrary that you think and need to be questioned too. You need to push against the assumption, hidden or denied, of the 'superiority' of the 'normal'. You need to support the equality of being 'different'. To do that you need to accept the dx.
The reality is that you cannot 'normalise' the DC on the spectrum, make them 'more normal', you can just make them to hide it for longer until they give up or explode. They do function differently and how differently does not depend on the parent but on the outside world, the pressure will continue to increase. It is the systems in society that "other" people on the spectrum, not the dx.
And any support and understanding at school and the DLA are based on the general awareness and recognition of ASD as a phenomena. If nobody was ever diagnosed, no one could have support with it.