They were having to be talked through filling in the form only to hit a roadblock when it came to proof of address. After expressing their voluble disbelief at some length that the handwritten doctor's note they had would not suffice, they eventually left empty-handed. (Incidentally, they were only looking the card to use it for ID for flying, they had no interest in voting).
Now these guys were obviously jokers, but it shows you will need a certain degree of application and time to get even the most rudimentary of verifiable ID. Even the conscientious may find themselves not getting around to getting the ID before election and losing their vote.
Why not have a national ID like in some European countries? Issue it at age of 15, you can go to a government office with parents when you are 15 and get your ID.
You can pass a sensible legislation for this. Have a grace period of 2 election cycles to allow all the people who want to vote enough time to get their ID (10 years is more than enough time to prove your address).
Presumably at least a few of them anticipate the easy problems and design a methodology appropriate to dealing with them.
For instance, if voter ID is highly effective, you'd expect much higher rates of double votes under a given registration in places that don't require it (unless the cheaters are masters of anticipating registrants that aren't going to vote).