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 invisibility ends if you attack or cast a spell.
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:
make it gesture, speak, and behave in whatever way you choose
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):
For the duration, you can cast spells as though you were in the illusion’s space, but you must use your own senses.