Does paralysis prevent actions that don’t require movement?

Question

The paralyzed condition states:

A paralyzed creature is incapacitated (see the condition) and can’t move or speak.

And for incapacitation:

An incapacitated creature can’t take actions or reactions.

Compare this to the 3.5 version, which specifically mentions physical actions:

This special attack renders the victim immobile. Paralyzed creatures cannot move, speak, or take any physical actions. The creature is rooted to the spot, frozen and helpless. Not even friends can move his limbs. He may take purely mental actions, such as casting a spell with no components.

Does this mean that, counterintuitively, by rules as written in 5th Edition a paralyzed creature cannot take actions that don’t require movement (e.g. casting a spell with no components)? It seems a silly interpretation, but that’s what the rules seem to be saying (assuming this resource accurately reproduces the rules).

Answer 1

Yes, a paralyzed creature cannot take actions, including the “Cast a Spell” action.

Note that bonus actions are also actions. As are legendary and lair actions.

Answer 2

Paralysis prevents the casting of spells

The incapacitated condition removes all the action options for spellcasting

The PHB says:

Most spells require a single action to cast, but some spells require a bonus action, a reaction, or much more time to cast.

So, spells can be cast with either an action, reaction, or bonus action.

The incapacitated condition prevents any action or reactions from being taken:

An incapacitated creature can’t take actions or reactions.

Which means the only other option left to cast a spell would be a bonus action. Unfortunately, the rules for the bonus action state:

[A]nything that deprives you of your ability to take actions also prevents you from taking a bonus action.

Which means that the incapacitated condition also prevents you from casting a bonus action spell.

There are no spell with no components

Even if you were able to get around having to use an action to cast the spell, there would still be the issue of components.

There are currently no spells that have no components, so you would have to move or speak to meet the component requirements for the spell you would want to cast.

Even were you able to get around the component requirements using a sorcerer's metamagic for example, the action economy discussed above will prevent a spell from being cast.