How do I turn my top-heavy headboard into a freestanding headboard?
The bed frame that you showed at the link (pictured below) is, as you have determined, not at all suitable for having a large headboard attached to it.

 

This frame is really meant to be used without any surrounding bed adornment or to be used as a mattress support inside of a full surround bed set. The link picture tells it all:



The side panels of this bed set adjoin to the headboard to make a nice solid support. 

I would never make a headboard by a bed free standing; especially one that is five feet high. The last thing you would ever want is the thing tipping over on you when you are sleeping in the bed or doing some active activity on or near the bed. 

One choice you have is to construct a full bed set. Alternatively you could DIY a support frame that sits directly on the floor and bolts to your head board at one end. The corners of this frame could be joined using pieces of angle iron screwed to the wood. The picture below could give you an idea to work with. Such a floor frame could be made shorter and narrower than the mattress so that it is kind of under the bed. You could even make it out of dimension 2x8 lumber and pad it with a fabric or upholstery material.