Firstly reading comprehension doesn't seem too strong on this thread is a bit strong given that the actual title of your thread is To think people who claim they never remember names are dicks?. Most people are responding to that claim as the OP wasn't particularly clear and a lot of us might pre-warn, it is entirely dependent on tone and clearly dependent on the recipients perception and general level of defensiveness/their own insecurities.
I am terrible with names, also very wary of saying 'it's nice to meet you' etc because I may well have met them before & they remember me but I have nothing, no memory of them at all - it's not arrogance, it is in part an ADHD brain and a long and multi-chaptered chaotic life.
I wouldn't actually say 'Oh I won't remember your name' in a dismissive way on meeting someone (for the first time or not), nor have I ever met anyone who has said like that (as obviously that would be weird and rude). But saying it to avoid awkwardness, and that would be exactly the same words but as I said, entirely dependent on tone and context.
Thing is, the whole not remembering thing is real. The only reason I can think anyone would say it out loud (other than stone cold indifference or arrogance) is because they want to get in there first and diffuse the potential awkwardness when someone comes along 5 minutes into your chat and you can't introduce 'no name' because it has already gone out of your head.
Or, for example when I'm feeling haggard or couldn't find my make up and have turned up at the school gates and it turns out it's the school fair and everyone is socialising but I look like a bag lady because I was late and working from home and thought I could get away with a grab and run pick up but now I'm stood in the playground looking like I slept in a bush that may have happened last week I tend to make some joke about it so that I've already said what everyone else is thinking. It could be that scenario, what is actually happening is that they are jumping in there with the 'I won't remember' before embarrassing themselves. More than likely if you are secure enough to realise it is their problem not yours and they are simply pointing out their failing rather than your potential lack of importance then it is fair enough.
Or they could just be dicks, there is always that possibility because there are a fuck ton of them about.