I'm not happy with 5E's condition list at all, for some of these very reasons.
The conditions themselves are mere bullet points of effects without context and not defined in and of themselves. Restrained is not explained well at all. Obviously my legs are restrained because my speed is 0. But I'm also taking disadvantage to attack rolls...why do I need my legs to shoot my crossbow? Maybe it means my legs are fully restrained and my arms are partially restrained--enough to penalize shooting a crossbow or dodging a fireball, but not enough to penalize me picking a lock or making a stealth check... It becomes even more confusing when it's compared with grappled, which is sort of like restrained but not quite.
Then there's the fact that they don't sync up with the magic system at all. There's not a single condition presented that interferes specifically with spellcasting, other than the conditions that prevent all actions, of course. In my mind, grappled, restrained, prone, frightened, poisoned and exhaustion should have some detrimental effect on spellcasting, or at least concentration. A character with 5 levels of exhaustion (next to death) has disadvantage on ability checks, attack rolls and saving throws; has their hit point maximum halved and their speed reduced to 0; and yet suffers no appreciable penalty to spellcasting or concentration. Hell, stick him on a tenser's floating disc and have him spam fireball until his slots run out and you've got a decent arcane main battle tank.
As for the original OP, and my adhoc solution to most of this, would be to use the Con-check concentration mechanic to cast spells in greatly impaired circumstances. On a failure, you lose your action, but don't expend the spell slot. I don't think completely limiting spellcasting while in a web is beyond the pale, though, especially for a character that escaped falling damage while falling unconscious into it.