Spark Transmitter with flyback driver and good arcs

A directional antenna is always preferable to omnidirectional. Both for transmitting and for receiving. But, as you say there may be factors that preclude using a directional antenna. For example, for portable (especially hand-held) use, a directional antenna may be to unwieldy, so a whip or "rubber duckie" antenna is typically used. Or if you don't know the direction of the communication, a directional antenna can be a liability.

If your mobile end knows what direction the fixed location end is, then you would benefit from at least using a directional antenna where you KNOW the direction.

A directional or omnidirectional antenna can be used with any type of antenna at the other end. Some VHF and UHF communication use polarization where the receiving antenna should be using the same polarization (vertical, horizontal) as the transmitting antenna

Slayer exciter circuits are by design very inefficient. It is nothing more then a simple resonant circuit operating in a lossy manner. To get bigger sparks you need something more like a Tesla coil operating as a class D amplifier. Adding more turns will add resistance and reduce the resonant Q factor of the circuit, eventually giving you even less voltage gains for exponentially increasing amounts of power.

I would suggest you spend some time reading up on Tesla coils. Basically you are switching FETs or IGBTs on and off, not partially driving them in a class A style like the slayer exciter does.

A small half bridge Tesla coil can get you 1-2 ft arks and cost under $50 to build if you keep it simple. That being said, if you are not aware of the dangers of working with high voltages, building a Tesla coil is a good way to toast yourself.