Does gifting items in TF2 to another player give them Premium status

Yes, it does. Buying (or being gifted) the Orange Box is one of the ways to get "premium status" and the "Proof of Purchase" hat in TF2. In fact, this is one of the original ways to do so.

I own the Orange Box, which I bought a long time ago, and I have both of these things, even though I've never made a purchase via the Mann Co. store. Others on the Steam Forums have reported that even after the game went F2P, this is still the case.

I've never experienced this specific case myself, but I do have 7 accounts I used for idling in the past (which have since more than paid for themselves). If the account in question has the "Proof of Purchase" hat that was given out to people that owned the game before it went Free2Play, you could bring your account to the attention of Valve as that is a dead giveaway that your account should be premium. If the account doesn't have that item, then the only other way I know of to get TF2 premium is to buy at least one item in the store (Minimum of $5 in the steam wallet). If what you did was buy/receive the orange box or TF2 back when it was sold and just held onto it until after the game went F2P, it may or may not be a fringe case they haven't handled properly. Regardless, I believe speaking to Valve support directly will be your only solution to this, as they are quite unlikely to allow user action to change premium status short of giving them more money. Best of luck!