Paladins are also proficient with lighter armor down to cloth, and doubtless trained in all of them when they were a squire. Although your explanation is nice, it doesn't actually make any sense because the DnD system doesn't make any sense, nor does it try to. The system simply doesn't support a paladin out of plate, hence why they're gifted it at level 1. Are there even rules on how long it takes to take the armor off anymore?
If your paladin was trained and skilled at using all levels of armour he'd have the relevant ability scores to reflect this IE you wouldn't dump dex or Int. He's trained in all armours sure, but he's not good at fighting in them unless the mechanics reflect this. I can play a ranger with 8 STR if I want to and I'm trained in the use of all melee weapons, but am I skilled with them? No I hit like a little girl. A paladin with NO Dex or Int is a Paladin that sucks at anticipating incoming attacks and at dodging. He is the tough as nails guy that lets his heavy armour do the defending work FOR him. He is not skilled at dodging, nor is he good at reading the flow of combat to avoid incoming attacks, he knows that his armour will protect him and relies on that. If you're playing a Paladin that has some combative sense and is actually skilled at avoiding attacks you would have placed some points in INT and/or DEX to reflect this. If your Paladin didn't, then no you aren't any better at not getting gutted than the wizard barring of course your vastly superior HP (which IS a measure of your combative know how btw).
Don't blame the system for your character decisions. The crippled wheelchair wizard is actually restrained since they're sitting down and can't go anywhere and thus suffer the appropriate penalties namely granting combat advantage and taking a -2 to attack rolls. (In my opinion since the crippled character is in essence chained to the chair) so isn't better at dodging and avoiding attacks than the Paladin without armour (especially since again, he has next to nothing for HP).
During the adventure the Pally could pick up a reach weapon to help with his lower AC, will use more healing ability on himself and can use Bolstering Strike constantly to give himself even more staying power round by round.
IMO the system does no armour just dandy. But if you build a character that so utterly depends on his armour, you're going to have to alter how you play him in order to take into account this new weakness. Just like a warlock is similarly hosed in a narrow dungeon or in difficult terrain (a sewer for example) where he can't get away from melee monsters and now can't even shift to get off any ranged attacks. Or how a party that has no wizard or torches is in for a rough time in a cave without any lightsources.
Paladin's have a lot of options, he could take up a bow and use Divine challenge and ranged attacks to force enemies to go through his allies to get to him or burn, best would be a heavy thrown weapon since he could use his higher STR for attack roles. I'm sure he could find something heavy to throw at them. This plus his encounter and healing abilities would make him a very useful party member when he wants to avoid getting hit. He could wait until the other strikers move up, then move himself into a flanking position and using divine challenge on that target in essence trading hits with his strikers to spread out the damage. Bolstering strike again grants the Paladin a kind of staying power the other characters don't enjoy as he shaves off 2-4 damage he would take in that round.
You could also build a CHA, INT, WIS Paladin intent on multiclassing into wizard for some ranged attacks and forgo heavy armour completely without hurting your much character at all. Half elf paladin like this could have the following at lvl 1:
STR 11
DEX 8
CON 14
INT 16
WIS 14
CHA 16
AC without armour: 13, 15 with heavy shield, with hide armour 16, with heavy shield & hide 18. Not to bad at all. He'd have a great range of other defenses as well, much higher reflex than your standard dump INT DEX paladin. You have 2 lower AC than a full plate Paladin but could take Thunderwave from Arcane Initiate to give you the ability to knock people back from you negating combat advantage and other attacks against you, or take Eyebite with your Dillante power to give you a round of invisibility against youre Divine Challenge target.
Saying the system doesn't support a Paladin out of heavy armour is being lazy because it simply isn't true. If you build a paladin that is gimped by his lack of heavy armour the class still affords you a slew of other options to remain a potent party member, you just have to change how you're playing a bit.