I think we should all just realize that Derek aka Veritasium is a youtube presenter, irrespective of his background or education and understanding he stands to make money from his youtube persona. Youtube due to it's algorithms which are tailored to the the preferences of the majority of viewers is not the best place for real science because in order to succeed it demands one to use speculative and sensational titles and similar video presentation. It is no secret that the channels that lately earn the most from that platform and are among the most popular do just that.
This being said I still find many Derek's videos entertaining and interesting and his science backing is still some of the most on point although it seems he himself deviates from that once in a while and this lamp video just happened to be one of them.
From a physics perspective there is no miracle here, i think
@Baluncore and others were right saying that due to transmission line theory the bulb will indeed receive some minor ever increasing current almost instantly while the real current necessary to light a real practical bulb will get there only when there is full load current established all around the length of the wires and this will happen at c along the wire in vacuum or less than c in other mediums.I think Derek was a bit tricky and deceptive on purpose making the wires long in one direction while making the separation short in the other direction which happens to be the direction/plane in which the voltage source/ switch and load is located.
If the battery was located at one of the far sides of the rectangular loop while the load at the other then the answer would be completely different and in fact then the bulb would receive it's full current at the time the first fields also reach it.
In a sense he made this demonstration somewhat similar to a card trick whereby the details of the setup determine the outcome not so much the physics behind it.