That Video Is "Clever"
Not actually clever, though. Just irony-quotes "clever".
The Initial Premise Fails
First, the fundamental premise that "every feature in the game has range of some sort when it comes to targeting anything," (verbatim from the video) is just not true. Animal handling, just two paragraphs above the medicine check, reads:
Animal Handling. When there is any question whether you can calm down a domesticated animal, keep a mount from getting spooked, or intuit an animal's intentions, the GM might call for a Wisdom (Animal Handling) check. You also make a Wisdom (Animal Handling) check to control your mount when you attempt a risky maneuver.
Only the last sentence deals with the PC's own mount during a maneuver. The first sentence applies to all domesticated animals, mounts, or just animals in general depending on the application. Does it mention a range? No. Does it mention being able to see the creatures? No.
In general, on skimming the entire section on skill checks, ranges are implied by the type of activity, not spelled out in exhausting detail. Just skimming that single page, activities requiring skill checks without specifying ranges or being able to see the target, which lead to the same sort of absurd situation, include:
- Communicating with a creature without words
- Winning a game of skill
- Estimating the value of a precious item
So that premise just fails. Maybe the youtuber is thinking of spells. Maybe they haven't read the rest of the skill section. Maybe-- just maybe-- it's not actually a good faith argument or misunderstanding at all.
DMG Quotes On The Role Of The GM
Second, if you really need textual quotes from the 5e source books to shut this down (which in my opinion you do not), try these, separately or in conjunction:
DMG 5e, p 5:
The rules don't account for every possible situation that might arise during a typical D&D session.... Sometimes mediating the rules means setting limits.
Takeaway: The rules are not and cannot be "complete" in the way this youtuber implies. Part of the GM's job is to fill in (presumably with common sense) when such situations arise.
DMG 5e, p 237:
To speed things up, assume that a character spending ten times the normal amount of time needed to complete a task automatically succeeds at that task. However, no amount of repeating the check allows a character to turn an impossible task into a successful one.
Takeaway: Some ability checks are impossible.
DMG 5e, p 238:
It's your job to establish the Difficulty Class for an ability check or a saving throw when a rule or an adventure doesn't give you one. Sometimes you'll even want to change such established DCs.
Takeaway: The GM is not bound to honor established DCs when they don't make sense. The GM can change them based on the situation.
Overall
No, that argument is not correct. Just, no. A GM is well within their rights to rule long distance medicine checks as impossible.