I'm strongly opposed to having prerequisites for any type of character option, unless the option is literally useless or nonsensical without the prereq (e.g., to take a feat that improves your spellcasting, you must be a spellcaster).
I dislike a game that encourages builds, and prereqs are a major culprit in that; they force you to plan your character out from the start instead of developing over time. These prereqs are especially bad because you can, with the right build, evade the prereq! Let's say you want to play a high-Dex, low-Strength rogue, and dip fighter for Two-Weapon Fighting style and Second Wind. If you built your rogue and played for a while, and then decided to detour into fighter, you're out of luck. The only way you can qualify is if you're willing to waste 2-3 feat slots pumping your Strength to 15, which is a horrendous waste of feats and will take many levels to accomplish. But if you planned the character from the start, you can put your first level in fighter and then multiclass rogue for the rest, and everything's cool. You even get two extra hit points out of it.
There is no good mechanical reason for these prerequisites that I can see. As I said, you can mostly evade them by planning your build in advance, so the balance impact is negligible. They don't "protect you from yourself" to any great extent--you can still end up with a crappy mix of classes through multiclassing even if you meet the requirements. And as for verisimilitude, multiclassing is
more plausible to me than single-class advancement. When you start learning a new thing, you tend to pick up the basics fast. Getting better at something you already know pretty well is much harder. When somebody explains to me how a wizard can go from "apprentice with a couple of spells" to "earth-shattering archmage" in a matter of months, or how a fighter can go from "greenhorn" to "slayer of armies" in the same timespan, then I'll be willing to discuss the realism of multiclassing.
That's a problem with D&D stats in a general sense, though, isn't it? I can't make a clumsy thief, gullible cleric, or weak barbarian without sacrificing the character's effectiveness. In general, not just here.
Sure you can. Wisdom is only important to a cleric for save DCs and certain Channel Divinity options. You can play a cleric of the Life domain and use your spells for buffing and healing, and your Wisdom won't matter at all. Dexterity is important for some of a rogue's class abilities, but most of those abilities are not critically important ones; I can see how to make a very effective rogue focusing on Strength and Charisma. And a dual-wielding Dex-focused barbarian is arguably
better than a Strength-focused one, since you can wield finesse weapons, double up on your rage bonus, and stack your Dex with your Con for Thick Hide.