The 5e rules have already destroyed this class concept (such as it is!).
Not at all. There is no rule in 5e that says what character creation options are available for a given campaign.
So, the game adding more options = the game taking away options?
That's not the best way to word it. Mechanics and concepts map to one another. Adding mechanical options usually just changes some of the mappings (but not always).
So you can keep the concept of strongest and hardest to physically kill. It's just that concept maps to a barbarian now.
Likewise you have eliminated the concept of strongest and hardest to physically kill from the fighter class due to it's mechanics no longer respresenting that in the game world.
As such you have created new and interesting mechanics with the barbarian class with the result of lowering the number of concepts the fighter class mapped to.
The DM gathers the players for the start of a new campaign, and says to make a 1st level PC using the PHB, point-buy only. One of the players had previously downloaded the free rules, and come up with a concept of 'strongest starting human', which is Str 16. Are none of the other players allowed to use the PHB classes except what that player knows? Are they not allowed to have a PC with 16 or 17 strength, in case it 'upsets' that one player?
If the group has agreed to use the PHB point buy only then that player upon explanation would realize his error and instead keep his character and modify his class concept or keep his class concept and change to the barbarian class. Either would be acceptable. In the situation you describe the game rules were already set out and there was a misunderstanding. Since he was the one in error he needs to be the one to adapt and make the game work.
So player 1 dislikes multiclassing because they (irrationally, BTW!) feels that someone else multiclassing takes away their choice to play their concept.
I've already elaborated why it's a perfectly rational position. Calling it irrational now serves no purpose other than to inflame.
But surely player 2 would have their concept actually taken away by player 1's objection.
Actually... player two can most likely keep their concept they would just have to map it appropriately in a game without whatever class they were originally going to take. What they couldn't have is whatever mechanics that class provided to go along with that concept.
Kind of like how player 1 can keep his concept and change classes to one that better suits his concept if more are allowed but won't get to have whatever set of mechanics his original class choice provided to go along with the concept.
Why does player 1's irrational preference trump player 2's rational choice to play what they want?
Player 1's preference is the same as player 2's. They both want to play a specific character concept with a specific set of mechanics.
(Yes I know I've slightly changed definition of concept in this elaboration as I didn't really have a better word. Concept definition 1 = class and identity intertwined. Concept definition 2 = identity apart from class.)