The difference between the filters you name is not that each new one invented made a closer approximation to the ideal filter, but that each one optimizes the filter for a different characteristic. Because there's a trade-off between different characteristics, each one chooses a different way to make this trade-off.
Like Andy said, the Butterworth filter has maximal flatness in the passband. And the Chebychev filter has the fastest roll-off between the passband and stop-band, at the cost of ripple in the passband.
The Elliptic filter (Cauer filter) parameterizes the balance between pass-band and stop-band ripple, with the fastest possible roll-off given the chosen ripple characteristics.
Now if I was to take my 5th order structure and was able to simulate for every possible inductor value and capacitor value would I find a combination that would give me the best possible / closest model to ideal, that beats all previously known filter types?
It depends what you mean by "best possible" or "closest model". If you mean the one with the flattest response in the pass-band, you'd end up with the Butterworth filter. If you mean the best possible roll-off given a fixed ripple in the pass-band, you'd end up with the Chebychev design, etc.
If you chose some other criterion to optimize (like mean-square error between the filter characteristic and the boxcar ideal, for example), you could end up with a different design.
Do mathematicians / engineers know of a "best" filter response that is physically possible for a given order but so far do not know how to create it.
The filters you named (Butterworth, Chebychev, Cauer) are the best, for the different definitions of "best" that define those filters.
If you had some other definition of "best" in mind, you could certainly design a filter to optimize that, with existing technology. Andy's answer names a couple of other criteria and the filters that optimize them, for example.
Let me add one other question you might ask as a follow up,
Why don't we in practice design filters to optimize the mean-square error between the filter characteristic and the boxcar ideal?
Probably because the mean-square error doesn't capture well the design-impact of
"errors" in the pass-band and stop-band response. Because the ideal response has 0 magnitude in the stop-band it's hard to define a "relative response" measurement that has equal weight in both regions.
For example, in some designs an error of -40 dB (.01 V/V) relative to the ideal 0 V/V response in the stop-band would be much worse than an error of 0.01 V/V in the passband.
The 2nd design is an inductive coupled example using an AM radio. The other two pick up an alternating electric field just like your body picks up AC alternating electric fields as discovered when you touch the input of a speaker amplifier and you get a low hum sound through your speaker accompanied by higher frequencies if you are in the vicinity of other circuits such as switch mode power supplies.
For a simple AC power example, the electric field lines exit a power conductor tangentially and "connect" or "end" at anything that represents an "earth". Normally these field lines are quite short because the neutral wire in AC power wiring represents "earth".
However, you can make a higher frequency oscillator that puts a signal onto a plate/wire and the capacitive connection that the person holding the transmitter makes with ground will cause electric fields to emanate. The receiver, likewise has a plate/probe/wire and can intercept these fields. It has an amplifier to boost the small signal it picks up and this is heard thru the speaker.
Both transmitter and receiver rely on a person or large object making a capacitive ground connection unless the equipment is powered from AC and earthed already.
That's my take on it anyway.
Best Answer
An active filter, i.e. which contains a power source, i.e. an op-amp, can be inductor-free. This is important for audio circuits where the frequencies are low and the inductors are huge and lossy due to their internal resistance. Consider the design for a 20Hz 4-th order filter and it can be either $2 of op-amp and capacitors fitting in 20x20mm, or it will be hundreds of dollars of inductors that take up perhaps a few litres of space and being horribly inefficient due to parasitic resistance in the inductors.
For radio frequencies however, the frequencies are often much greater than the bandwidth of any op-amp that you can buy. So while theoretically you could build an inductor-free filter using an ideal op-amp, there is no such thing as an ideal op-amp. You can't buy an op-amp with 100GHz of gain-bandwidth product! In other words, once you get up to higher frequencies, the op-amp has no gain any more and your filter won't work.
Happily though at radio frequencies, the inductors can be really really small. And by small, I mean surface mount and about a cubic millimetre of volume. So all the drawbacks of inductors at low frequencies are no longer relevant, and there is no real need to build inductor-free filters.
Radio filters can also implemented using tracks on printed circuit boards, which are themselves inductive. By using careful PCB track placement, you can create a microstrip filter which uses the parasitic inductance of the track and the inter-track capacitance to create your desired filter.