It's connected to the optic nerve and is a genuine eye.
If it acted normally Gaara wouldn't need to close his eye in order to use it, but he does. And if it is connected with chakra to the optic nerve then yes, it can in fact be linked to the brain via the 3rd eye. Stop making it seem like it works through wifi or some shit.
Your description of the jutsu says it all.
"sand made eye". That's all you need to know, no need to delve further. As for your "wifi" statement, it's been shown in the manga before [
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] so I fail to see how it's hard to believe. Gaara shuts his eye simply because the optic nerve for that eye is now inactive and he transfers it's concentration into the sand based one. Optic nerves transmit visual information to the brain, so Gaara closing his eye would logically mean that while third eye is in use, the shut eye is inactive due to the sand eye needing the optic link. The eye is pure sand, no optic nerves are present inside of it so your premise is more or less impossible.