Two gloomstalkers slowly move through the Underdark silent as ghosts. Their prey, a band of drow led by an inquisitor, are completely unaware of their presence. Suddenly, the rangers attack quickly followed by the rest of their party. One ranger restrains the inquisitor with an ensnaring strike, and the party taking full advantage of their surprise quickly reduces the drow band by half in one round, but then the next round starts, and the inquisitor takes her turn. She starts chanting while performing forceful and/or intricate gestures in order to cast a spell while the vines that bind her hinder her gestures... in no way at all.

By RAW, this is a perfectly reasonable action. The restrained condition doesn't say anything about limiting a person's physical actions beyond reducing there speed to 0, making it harder for them to hit with any kind of attack, or dodge out of the way of damage, but it seems like should doesn't it? A creature unable to move and attack freely should not be able to freely cast spells that require somatic components. As things stand now, somatic components are completely unpreventable without incapacitating the caster. You can gag or silence a caster to prevent their use of verbal components; you can take their material components away from them, but knocking them out (among other effects) is the only sure way to prevent them from waving their hands around

What do you all think? Should a creature be able to use somatic components while restrained at all, should they have to succeed on a Dexterity check for the spell to actually go off, or am I blowing this all out of proportion?

Speaking about blowing things out of proportion, after the above encounter, my ranger started breaking the fingers of every prisoner he took. His response to the rest of the party when questioned about this new habit was "It's not like tying them up really does anything."

I think it'll depend on the exact circumstances and how you interpret certain spells and effects, and the description of the somatic components (if there is one).

For example, entangling vines slow your movement, so the implication is that they're only really entangling your legs, leaving your hands free to take actions as normal. If you were to be knocked prone in an area of entangling vines however I'd say that's a different situation, and probably rule that the arms are bound as well. If the somatic components require arm movement, or to point at or touch anything specific, then you probably couldn't do it in that case. Same with a web; if you run headlong into a giant spider's web you're probably going to end with both your arms and legs, body and face stuck in position, requiring a strength check to free just a single limb.

If a creature were to grapple a spellcaster with the aim of preventing them from casting, then I'd probably rule that either the grappler can simply do that (no somatic components allowed if grappled) or can use their reaction to force a contested check to prevent casting, e.g- if the caster tries to move a hand, grab that hand, but in that case you might also allow it for verbal and material as well (hand over the mouth, block a component pouch etc.).

You're right that the rules on somatic components are a bit weak, but really that just means that it's up to the DM in each case.