I was looking at these cards and wondering if TGI Friday would counter Magnites, but I don't think it will. Am I correct?

My reasoning is as follows - Starhawk cuts to speed 0, TGI activates THEN. At the end of the Starhawk's turn it activates Magnites which then cut the speed of the ship TGI is riding on.

Yep. The Grand Inquisitor can help a bit, as it would give you a free way to speed up afterwards, but you probably won't get a chance to before the Starhawk chews you up. And he's taking up your officer slot.

TGI can only resolve when an enemy ship changes its speed.

The Grand Inquisitor
"When an enemy ship at distance 1-5 changes its speed, you may exhaust this card to increase or decrease your speed by 1."

Magnite Crystal Tractor Beam Array

"After the end of your activation, you may exhaust this card to choose 1 enemy ship at distance 1-5. That ship's speed is increased or decreased to match your speed.

While your speed is 0, you cannot ready this card."

Meaning TGI only resolves during an enemy ships maneuver (in this case the starhawk) but MCTBA takes place "after" that ship (the starhawk) has finished activating. Meaning TGI wouldn't be able to be resolved as the enemy ship isn't changing its speed.

So, TGI is sitting on your Demolisher (current speed: 2). The enemy Starhawk, out at distance 3-ish, activates, shoots, then drops its speed from 1 to zero during its maneuver.

TGI works *now*. You exhaust him to increase Demolisher’s speed to 3.

Then the opponent triggers his Magnite Tractor Beam, targeting Demolisher. Demo is forced to match speed with the Starhawk, and drops to zero.

The only time TGI would offer any protection against Magnite Tractor Beams is if you hadn’t exhausted TGI, the opponent hit Demolisher with the Magnites, and then (foolishly), he decided to change the speed of another one of his ships later in the turn. TGI could trigger then (as long as he’s in range), allowing you to escape the trap.