Question: Would using GPS antenna that only points at the sky (vs more omni-directional ones that mobile devices must have to work in any position) would "fix" the jamming problem ?
On paper. In reality, most every antenna has "side lobes" through which jamming signals can spoil the actual signal. In some cases the more directional an antenna becomes, the more significant are the side lobes.
It is in the linked-to article, including discussion about lobes, polarization, more directional replacement antennas, and comparative measurements while being jammed.
I can't find it in a quick Google, but I read an article a while back talking about them putting the receivers in foxholes; can see the sky just fine, but any ground-based jammers nearby have to push through the dirt.
Remember that GPS satellites may be near the horizon relative to you. That said there are enough of them in orbit that you can usually get away with only using the ones that are more overhead.
On paper. In reality, most every antenna has "side lobes" through which jamming signals can spoil the actual signal. In some cases the more directional an antenna becomes, the more significant are the side lobes.