Why do most phono preamps lack XLR input even thought cartridges are naturally balanced?

Most phono stages, preamp and amplifiers aren’t true balanced. A balanced input doesn’t necessarily mean it’s balanced. My Atma-sphere  mp-1 is fully balanced with XLR phono input. I had Ralph add rca phono inputs so I could use any tonearm without adapter. Not many arms have  xlr connectors . All the arms I’ve ended up with over the years except one has been rca. 

With a cartridge, you only have a ground connection on one end. Equal currents induced in the same direction on each wire (common mode) "induce" the same voltage and cancel each other out (it is a loop), at least at practical audio frequencies. Differential noise, will of course look just like a signal. The cartridge is already "isolated".

That is true, as long as currents flow in the loop only.  If one of the wires go to input while the other goes to GND (unbalanced input), currents in both wires are not even anymore.  Perhaps, that's why floating (for audio frequencies) balanced output, like transformer is not enough and has to be connected to balanced input (not single ended).  Microphones also benefit from balanced input.  

As for Instrumentation amps, yes they have laser trimmed resistors, but front (two amp) differential section has always gain of 1 for common mode signal independently of resistor tolerance.  If you set gain of 100 for this section you get automatically 40dB CMRR independent of resistors tolerance.  It happens because each of two amps is referenced to input of another (instead of the GND). 

https://www.researchgate.net/figure/Fig-Two-Op-amp-Instrumentation-Amplifier_fig1_299514235

That is the only way I can see, for truly balanced amps, to function without converting common mode noise to normal mode signal.  Otherwise matching resistors and keeping them matched to some sensible number (like 60dB=0.1%) is not practical, while adding RIAA frequency correction and matching capacitors to 0.1% is next to impossible.

Thanks for the mention, Kijanki. To add further technical elaboration to this erudite discussion of esoteric matters, IMO the main reason "most phono preamps lack XLR input even though cartridges are naturally balanced" (quoting from the subject line of this thread) is simply what Lew said earlier:

... it’s a longstanding tradition to offer only RCA inputs.

:-)

And of course traditions often tend to be self-perpetuating.

It’s perhaps also relevant that not too many decades ago I believe a considerable majority of high quality turntables had pendant (non-detachable) phono cables terminated with RCA plugs.

Finally, regarding RIAA equalization Ralph’s (Atmasphere’s) MP-1 and MP-3 preamps are of course fully balanced, provide transformerless balanced XLR inputs for their built-in phono stages (RCAs can be added as an option), and are spec’d as having RIAA accuracies of 0.07 db and 0.1 db respectively! And given especially that it is Ralph who has provided those specs, I have no reason to doubt them. I’m not in a position to elaborate on how he accomplishes those numbers, of course, other than pointing out that he uses triode-based differential stages rather than separate signal paths for the two legs. But as far as purely technical considerations are concerned his designs certainly speak to the practicability of accomplishing what the OP has asked about.

Best regards,
-- Al

one would have to introduce a true balanced phono circuit internally.

While the idea of duplicated, bridged, or fully balanced (whichever term you like) is sexy, the idea that this is the only advantage is really not correct.  Professional gear has been using balanced signals long before audiophiles got it into their heads to duplicate circuits.

Having a differential input which is not referenced to ground has a lot of benefits for noise.  And while I don't think we need full-sized XLR plugs, there are mini-XLR plugs which would work a treat for this.  I've often wondered this myself.

Phono cartridges are floating sources rather than balanced, and unless there is a low-impedance connection between phono amplifier ground and both sides of the cartridge signal coil, and/or the common mode rejection of the phono stage input stage is extremely high across a wide bandwidth, there most likely will be substantial pickup of electrical noise from the environment (i.e., connecting the phono cartridge in balanced mode will probably yield worse noise performance than an unbalanced connection).

An input transformer magnetically isolates common mode noise, and is a good solution to building a quiet balanced phono stage.