Find the Q-factor for coils and understand weak power transfer
Any tips or resources would help.

The Wurth coil as sold by Farnell: -

What do you think that the black support sheet is made from?
Answer - it's partially made of ferrite material and, it will highly concentrate the fields from your transmit coil and collect more energy in effect. If you want to prove this, get a magnet and feel the attraction force. It should be maximized at around the centre of the plate i.e. the ferrite is probably a disc placed centrally to attract the AC magnetic fields through the centre of the coil arrangement. You want magnetic flux to be drawn through the centre.
Here's what Farnell say about this coil: -

The 760308111 is a wireless power charging Transmitter Coil with pot
core results in limiting the stray field in the design. This WE-WPCC
coil features high permeability shielding coil

It's not great or perfect language but, I think you can see that it uses ferrite material to enhance the ability of the basic coil. The data sheet also says this: -

Direct mechanical impact to the product shall be prevented as the
ferrite material of the core could flake or in the worst case it could
break

Notice the words core and ferrite!
Your coil (this one): -

It has no ferrite material to enhance its ability to collect energy - that's your first problem and it's a show-stopper in my opinion. Of course, if your receive coil and transmit coil were equal size then it's less of a problem.
The second problem is that it only has 3 turns and is therefore not resonant at the same low frequency compared to the Wurth coil.
If you then made your transmit coil resonant at the higher frequency (to make it compatible with the resonant frequency of your homespun receive coil), there will be less current in the transmit coil because its impedance will be higher. Therefore less energy transferred. OK, if you are series resonating the transmit coil this may be less of a problem.
Given that this type of arrangement is a transformer, having fewer turns on the secondary also means less voltage produced by the receive coil.
In all of these types of experiment, you should be prepared to re-tune with capacitance and tweak the driving frequency and keep checking this. You only really need to look for voltage amplitude on the output coil so trying to measure S parameters is missing the direct point. Use a signal generator and use an oscilloscope to set it up, then use your other equipment to make precise measurements.
I might add that in all the power transfer coils I've built I only ever used a signal generator and oscilloscope and I got great results and built probably about ten systems for customers that are still in use today.