I have to agree Alex, as an almost 30 year controller in various countries I don't find alot wrong with the controller beyond the miss of the readback of the heading.
That clearly was a big miss and can't happen. That said also I recall an incident I was involved in over 20 years ago when a Spanair plane I was working I gave a heading of 150 to and he climbed to FL150 into conflict with other traffic. We listened to the tapes afterwards and even with 6 of us listening and playing it back over 10 times knowing what happened none of us heard the word heading due to the pilot's accent. We all heard 150 but heading/FL was completely unintelligible.
It's all very well listening to these YouTube videos but nobody knows exactly what was heard on the headset at the time.
As to the conversation after really I don't get the criticism. It's impossible to make judgements based off these YT videos because we have no idea what was edited out, and if teimescales are compressed or not. IF the radio calls we here are all in real time with nothing edited out the frequency wasn't busy. The United pilot was given a brief explanation of what happened in case he needed to file a report, which he then confirmed he did. Then tje controller provided further information for the United crew which they seemed to appreciate while he continued to control the traffic effectively.
58DH was only ever cleared to 6,000 multiple times which was read back correctly by the crew, there was absolutely zero reason for them to ever start climbing, the controller didn't need to keep repeating that over and over because it already been passed and acknowledged several times already.
The 58DH crew sound like an incident waiting to happen with absolutely no attention being paid and zero situational awareness.
Controllers don't have time to play babysitter, professional pilots are expected to be exactly that and be competent enough to follow basic instructions and to pay attention when operating in complex, high density airspace.