Can you use the Government Travel Card to pay for a rental truck to haul HHG?

Yes that's what it's for. Now that finance no longer does advances the GTC is supposed to be used for PCS expenses such as tickets or rental trucks.

The thing that's missing is your orders should have a statement/endorsement saying in effect "Use of GTCC authorized for blah blah blah". There's typically standard statements that get dropped into the orders on this and other allowances or restrictions like the "not to exceed" stuff. If you don't see it, the S-1 is being lazy. These endorsements help keep you and the bean counters on the same page.

Is it a PCS expense? Yes, since the army is asking you to move and is therefore a Business related expense...I say this with A LOT of confidence as I am a Finance NCO and have been finance for 15 years