If a FM Radio station broadcasts live performance/concert and if we listen using a good FM Radio then is the whole chain analog? I mean I don't see the need for ADC and DAC conversion
Sorry, it's not going to be analog the whole chain. In modern FM transmission chains it's far more likely there's been a conversion to digits at some point early on. Audio processing, which every FM station must have, has been all digital for quite some time. The stereo transmission signal is digitally generated, and in modern FM transmitters, the entire modulated carrier is generated digitally. Going backwards in the signal chain, the program is most likely being mixed on a digital mixing desk, and if the concert takes place in a concert hall, the final mix will be transmitted to the studio digitally, where it will enter an digital audio infrastructure for switching, routing, and mixing, then head off to the transmitter as a digital signal.
In that case, this should sound more natural right? or am I missing something?
You're missing the fact that handling a signal digitally from mixing to transmission is far, far less subject to damage than the old analog method. There's no reason an entire analog path would sound more natural, never has been.
The chain you're listening is all analog. Most likely the source is digital and is run through a digital broadcast mixer (DAC) then broadcasted as FM radio waves
Most of the FM transmission chain is now digital up to and including the final modulated carrier. It's been this way for some time now, with benefits both in performance and stability.
But if its a live performance then why the source should be digital? Can't they record analog form and broadcast directly?
If it's a live performance, then it's broadcast without recording of any kind. Microphone signals are most likely mixed together using a digital mixing desk, and after the desk there's no reason not to handle the signal in the digital domain.
Digital mixing desks have many advantages in live broadcasting, not the least of which are stored and recallable settings and automation. Live concerts are rehearsed, and broadcasters will attend and mix the rehearsal, making notes of settings, and storing them for recall during a live performance. Digital mixing is done at 32 bit floating point internal processing resolution, if not 64 bit FP.
I believe the original question is rooted in the concept that analog signals are somehow more pure than digitally handled ones. Probably, the OP thinks that digital signals are made up of tiny digital "steps" forming a "staircase" type of waveform. This is simply not true, and never has been true. There are no steps in the output waveforms from a DAC, none at all. Because an analog signal can (and does) pick up analog noises and distortions along the way, it's been more difficult to handle a total analog signal from microphone to transmitter without damage. This is particularly true for live concert broadcasts were the mixed signal must be sent some distance to a radio studio or broadcast facility before being transmitted over the air. Before digital technology, the method was to use copper wire pairs belonging to the telephone system. Wire pairs longer than a thousand feet or so required equalization to maintain flat frequency response up to 15kHz (forget 20kHz). The longer the run, the more equalization required to restore the high end. The more equalization, the greater the noise, and the longer the run, the greater the noise pickup. It was kind of a loosing battle until good high speed digital transmission became possible. Now we can send perfectly flat 24 bit digital audio over an Internet connection anywhere on Earth that has adequate network speed, with no change in frequency response, distortion, channel separation, or noise. The audible difference is not small.
Also be aware that the transmission of a stereo signal on FM involves a few processes that are far more difficult to pull off well in the analog domain. FM Stereo handles Left and Right by transmitting the sum of the two (L+R) as a mono signal with response up to 15kHz, then transmits the difference of the two (L-R) using a modified AM system (double-sideband suppressed carrier method) with the carrier centered at 38kHz, but suppressed. The carrier is re-injected in the receiver. The L-R subchannel is far more noise-prone than the L+R signal, resulting in noise, distortion, and high frequency separation loss. Modern transmitters (actually, audio processors) develop the entire FM baseband signal comprised of L+R, 38kHz L-R, and 19kHz "pilot" signal digitally, which results in much higher separation and lower distortion. Today, all of the transmission quality limitations exist within the receiver, which is most likely a digitally controlled analog demodulator.
Please try to get your head around the idea that digital audio signals are the "pure" ones, analog audio is the type that suffers loss and distortion along the way.