The role labels are meant to apply at the level of game play, not within the fiction. 4e doesn't hide from the fact that it is a set of rules for playing a game.
One conceit of that game is that enemies are typically encountered in relatively discrete groups rather than essentially endless waves. This is a conceit that D&D borrows from fantasy fiction, and particularly pulp fantasy. (The same conceit is found in other fiction with pulp origins, such as super hero comics. It can also be found in some non-puplish fantasy, eg romantic fantasy, because the characters regard it as dishonourable to deploy excessive force against an honourable foe.)
Given this conceit, there are only a finite amount of attacks to be delivered, and hence a finite amount of damage to be taken, before any given combat is over. Defenders "defend" because, in virtue of their mechanical abilities, they draw more of the attacks onto themselves then would result from a strictly random or even distribution of the attacks across the PCs, and thereby absorb more of the potential damage (via a combination of high AC meaning some potential damage is not actualised, and robust hit point totals which means that actualised damage isn't as likely to be fatal).
In the fiction, this mechanical "defending" may or may not take the form of defending. The game leaves it up to the player of the PC to colour what is going on, what exactly a mark means, why exactly the NPCs rush towards the fighter whose player has just used Come and Get It, etc. This is one of the points at which 4e is closer to a free-descriptor-style RPG (like FATE, or HeroWars/Quest, or Marvel Heroic RP) than other versions of D&D.
A "leader" may or may not lead in the fiction. (And p 16 of the PHB notes as much.) The leadership, from the role point of view, consists in the fact that they act as force mulitpliers for the party to which they belong - they multiply the force of the defender by supplying buffs and hit points, the force of the striker by supplying extra attacks, and of the whole party by conferring positional advantage through movement abilities.
A "striker", as you note, is not unique in attacking. What is distinctive about a 4e striker is that damage-delivery is his/her main combat function. The closest thing in the PHB to a pure striker is the archer ranger. (The melee ranger is more likely to have to find ways to withstand being at the centre of the scrum, and hence have at least a touch of defender capability; the rogue and even moreso the warlock mixes striking with control.)
A "controller", like a striker, degrades the enemy, but generally does so outside the hit point system. (In 4e, especially the PHB, there is a legacy treatment of AoE damage as also a controller function. This is entirely a legacy matter, because the classic D&D wizard both does AoE hit point damage via the artillery function, plus bypasses hit points via charm, web, slow etc.) The degrading takes the form of disrupting formations/positions (eg forced movement) and condition imposition (which often takes the form of action denial).
"Striker" and "controller" are not roles that occur in nature, as it were. They depend upon a mechanical convention, long part of D&D, that some ways of degrading an enemy, but not all, are measured via hit point depletion. In an RPG system in which there was no difference between hitting some one for "damage", and hitting someone and "stunning" them, the contrast between striker and controller would disappear (examples of such RPGs include HeroWars/Quest, Marvel Heroic RP, and arguably Tunnels & Trolls).
Certain features of controlling, defending and leading also depend upon the fact that D&D uses figures on maps (whether literally or in the imagination) to adjudicate combat positioning. In an RPG in which positioning was not treated in this way, forced movement and "leading" by way of conferring bonus movement would disappear, and hence some of 4e's role distinctions would also not apply to that game. Marvel Heroic, HeroQuest Revised and Tunnels & Trolls are all examples of this, and so is AD&D as far as melee (but not ranged combat and closing to melee) are concerned.
This is why I have stressed upthread that roles are about function in the context of fiction meeting mechanics. Strip away the action economy, the positioning rules, the distinction between hit points and condition imposition, and the 4e roles disappear with them.
But I don't think 5e has stripped many of those things away. It still has positioning rules based on a real or imagined map (eg movement rates, distances, effects etc all measured in feet). It still has conditions as a way to degrade the enemy without depleting hit points. It still has an action economy. So it still has the mechanical raw material out of which the 4e roles emerged.
One thing that is different about 5e, though, is the mechanical duration of combats. Fewer round of combat mean fewer events of character movement (either PC or NPC/monster), less time for imposed conditions to matter, etc. By making movement more liberal it also makes positioning in combat easier, which reduces the distinctive significance of granting allies free/bonus movement or imposing forced movement on enemies. These features have the potential to reduce the significance, in 5e, of the mechanical features that underpin 4e's roles - much as in AD&D, because melee is sticky by default and there is no movement once in melee, there is no distinction between melee strikers and defenders, nor any significant scope for leading or controlling that focuses on movement.
