Configure the first gain stage for high gain to maximise the SNR.
The TL084 is very noisy. 37nV/√Hz! They only still exist because people buy them by mistake.
So 70's! Choose something more contemporary say, OPA1664.
LC filters need controlled impedances to work effectively. 1k drive and 10k receive puts you in the op-amp's comfort zone.
Deal with the filtering in one filter stage where, that another useful bit of noise and DC error reduction.
Design a proper band-pass filter.
markimicrowave.com/technical-resources/tools/lc-filter-design-tool/

If you want to DC couple the non-inverting stages a cap to ground is needed, C1 in my diagram. It sets the corner freq of the lowest freq of interest. ie high pass.
That makes the amplifier stages unity gain at DC so that op-amp offset voltages are not amplified.
You only need AC gain.
Inverting amplifier stages will work just as well if not better. AC coupling is not an issue in this project.
The MC34063 is operating just an octave above your coil frequency.
That makes it very hard to filter out. unshielded DC-DC converter inductors radiate EMI. Bad.
Use a DC-DC converter that runs in the in or up near the MHz eg a TPS54202
And try this tool
https://webench.ti.com/power-designer/ read the datasheet and if you can lay it out like they say.
fbp.jpg (414.16 kB. 1376x1289 - viewed 41 times.)
Or maybe cut your losses (not the power ones though) and just settle for a linear regulator.
I doubt it needs to be precise. How sensitive is your oscillator to its power supply?
Do small changes in frequency matter considering component value tolerances in the resonant circuits?
Poor PCB layout and wiring have a negative impact on noise both conducted and radiated
You need shielded cables for your coils. make sure to take into account the cable capacitance. So many pF per metre (or fahrenheit per cubit in freedom units).
On the receive side use a shielded twisted pair cable and ground tie shield to the amplifier input ground.
Do the same thing on the transmit side, tie its shield to the TX ground to stop radiating guff into to other instruments in your lab.
Shield the TX and RX coils, tie to their respective cable shields.
My understanding is that you need lock in detection and that the front end amplifier in these sensor systems measure the current induced in the coil, not the voltage across it.
I wish you good luck.