The Project Image spell allows the caster to control an illusion of themselves from a significant distance away. They can see and hear through the eyes and ears of the illusion. The illusion can gesture and speak.
Can the caster cast spells (without material components) originating from the illusion? (through gesturing and speaking with the illusion)
Mislead has a similar effect, so the answer will likely apply to that as well.
Best Answer
No
In order for us to have a common language with which to determine how things work, we must stick to the idea that spells only do what they say and neither Mislead or Project Image offer an ability to cast through the illusion.
In fact, if you try to cast with Mislead:
The spells dictate what you can and can't do and how they work, and neither have language regarding the ability to cast a spell through them.
Confusion around the description
I think you may be reading more into the following line:
This does not mean you can cast a spell, it only means what it says in that you can have it do any of those communicative actions.
Just being able to gesture and speak does not give something a Spellcasting ability. Only having a spellcasting ability will do that, and without language stating such, the illusion does not have that ability.
An example of what you are looking for
As David Coffron points out, the Trickery Domain Cleric does have language in their Channel Divinity Invoke Duplicity(PHB, 63):