There are 2 exceptions to your generalization:
A minimum phase filter and a zero phase filter with the same amplitude response.
It is also possible to have an LC ladder filter with a zero gain amplitude response but shift phase in the region of interest over two decades
Minimum phase means the energy is front loaded or causal like a step input with a fast response and there is no energy before time=0 It can be applied to impulses steps and wavelets.
The minimal phase filter means the the energy rises quickly
Zero Phase means there is maximum energy at relative time at t=0
Maximum phase means the energy is back loaded like a Tsunami.
You cannot For smoothest linear phase shift or flattest group delay, always choose a Bessel filter. THis has low Q =0.5 For higher order filter ALLPASS filters with high group delay, the number of stages must be increased proportional to the delays. Q's are adjusted for each stage so that interstage effects result in a desired group delay passband response or in LC filters, impedance ratios in a ladder results in a flat group delay.
All pass Filter ( delay line)
LC All Pass filters are used to create delay lines for HF pulses or data or analog scope signals so triggers can be viewed.
LC Filters are low loss passive filters of any order and many topologies suitable usually for >100kHz where Op Amp gains are diminished or for passive DC power SMPS filters at lower ranges using the driving point impedance.
If you understand Active filters then you can understand phase and amplitude responses of LC filters. They can share similar characteristics but have unique advantages. With free software you can chose a design an active 7th order Chebyshev low pass filter and have a complete Schematic and Bill of Materials (BOM ) in less than a minute than it takes to read this answer, but with limitations of gain-bandwidths. RLC filters also exist for DC power LPF's used in Class "D" power Amps and SMPS ripple filters.
- For LC filters you must choose driving and load impedance.
- As in all design, first you choose the specs then and for filters you later choose topology and then realize it then repeat until specs match results with component tolerances, suitable impedance ranges, noise DC error, and cost.
- For active topology Sallen & Keys or multiple feedback or full differential
TI software allows you to later change to any standard tolerance values with a RC tolerance dropdown menu
This TI version does not balance input offset voltages for matched Rs from input bias current or allow easy scaling of RC, but the advanced user will know how to do this anyways. THe WEBENCH version is best to use as it is updated regularly.
Important reading on Filter types
http://www.ti.com/lit/an/sbfa002/sbfa002.pdf for a background on "some" of many kinds of filter properties used in both active RC and passive LC filters. In active filters impedance inversion is possible for negative resistance gain and negative reactance Capacitors to emulate capacitors but are limited by the gain bandwidth product.
For online login to www.ti.com ( free ) there are now hundreds of resources and free design software such as active filters http://www.ti.com/lsds/ti/analog/webench/webench-filters.page ( with TI logon)
You can also download the offline version but it is no longer updated http://www.ti.com/filterpro-dt to design the same Butterworth, Elliptical, Chebyshev
The TI software guides you easily to make any Active filter.
As always you must learn to define the Design Specs 1st with characteristics; (lowpass, highpass, allpass, bandpass, bandstop, (LPF,HPF,APF,BPF,BSF) ; parameters for gain, f1 breakpoint, passband gain ripple (error), f2 bandstop reference and attentuation