Yes extra inductance connected to the LC tank circuit becomes part of it, but not when its on the other side of a full bridge rectifier. The diodes prevent the energy from freely flowing back into the LC tank.
How do you imagine "diodes prevent the energy from freely flowing back into the LC tank", diodes are part of the LC tank and energy is perfectly free to flow between L and C both ways (only for L it is always in the same direction).

You could skip the rectifier and connect it directly in parallel to the LC tank circuit. But this adds a lot of extra energy storage to the tank so its resonant frequency will drop below 10KHz. You can partly mitigate this by making the capacitor in the tank circuit very small in value in order to get the resonant frequency back up. But this also reduces the current in the tank circuit at a given voltage level. So to still get your 8A in such a LC circuit you need to drive it at a higher voltage, again needing the 40kV across your coil, so as a result you need the DC supply to your ZVS circuit to be about 20kV and it must have transistors rated for 40kV.
"You could skip the rectifier and connect it directly in parallel to the LC tank circuit" i'm not sure what you mean here. I assume you mean the coil but coil is part of LC tank.
In any case there is no need for extreme voltages and power you guys mention since we are talking LC tank, no reactance here. This is why the ZVS heaters are so popular these days, they can push hundreds of amps at up to 200KHz (there are examples on YT). Zero voltage switching and no reactance, pure ohmic resistance and cold FETs.
I just calculated, with 0.18H inductor and 68nF cap resonance would be at 1.44KHz which is quite low. I will probably lower the number of turns/inductance to raise it to some degree.
There is no free lunch in physics. This is how much energy it takes to make such a strong magnetic field.
That is so wrong it's beyond belief. Truth is the exact opposite. I humbly suggest you to start learning about overunity devices, you are already 100+ years behind on actual developments. Here's a good book to start your research. It's only 2555 pages long, so loads of fun and enlightment ahead for you.
http://www.free-energy-info.com/PJKbook.pdf