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This question deals with flying by reference to old-fashioned "steam gauges", not a modern "glass cockpit".

I recall hearing some pilots very experienced (ATP) pilots advocate for the older-style "turn rate indicator" (senses rotation about the aircraft's up/down axis) over the newer-style "turn coordinator" (senses rotation around an axis canted about 35 degrees from vertical, to sense roll as well as yaw.) Specific reasons were given, but I don't recall them. In what ways is a turn rate indicator more useful than a turn coordinator, during "partial panel" flying (no working attitude indicator or heading indicator) in clouds?

The purpose of canting the gyro's axis of rotation to sense roll as well as yaw is to give an earlier warning to the first stages of a developing bank and turn, and in my experience this is indeed helpful, but apparently there are some drawbacks as well.

A turn coordinator's axis of rotation is tilted around 30 degrees and rotates in the opposite direction. Because of this, looping errror is reversed and it under-reads at more than 1g. This makes it essentially useless when recovering from an unusual position on instruments, which is one situation you can easily find yourself in when flying partial-panel.

With a turn rate indicator the instrument will over-read when the pilot is pulling excess G's in a turn; all the pilot has to do is relax the stick to correct.

You have the yaw/roll switched. The TC's canted gyro adds roll to the yaw rate sensing, not the other way around.

The biggest drawback to the turn coordinator was the presentation of the aircraft, which to someone new to its use can tend to get mentally processed as a horizon bar, especially an old timer trained on aircraft with war surplus black face artificial horizons (with were widely installed on light aircraft into the 60s until surplus stocks ran out). The TC's little airplane tilting left could initially be mentally interpreted as a bank to the right in other words, until your conscious brain finished processing the information (Russian artificial horizons had this presentation where the gyro tilted an airplane symbol; transitioning to western style horizons can be a problem for Russian pilots).

A turn and bank with the vertical needle is easier to interpret at a glance. The problem with the turn and bank was since only yaw rate information was presented, and there is no internal damping, in bumpy air you ended up with too much "noise" in the information presented from the nose wagging back and forth. In an airplanes like V tail Beech Bonanza, which were quite bad for tail wagging in turbulence, a regular turn and bank could be nearly useless in the bumps with the needle slewing back and forth constantly in level flight. TCs avoid this by not just splitting the motion sensing between roll and yaw, but also by using internal dampers (little shock absorbers) to present only movement that is established.

One problem with TCs is that once you are stabilized in a turn, there is no more roll rate to sense, only turn rate and that has to be taken into account in the way the rate markings are placed.

My old '68 Cardinal had a TC that had gone bad because the internal damper had gone south. The thing jumped around like crazy because of the un-damped sensitivity to both roll and yaw. I replaced it with an older 12v electric turn and bank (way cheaper - Turn Coordinators are quite expensive because they are electric with internal inverters to convert 12vdc to 110v/400 Hz fixed frequency ac for the gyro) and found I had a strong preference for the vertical needle presentation.

Personally, I think the turn coordinator should have stuck with the vertical turn needle presentation, but with the damped canted gyro internals. Like I said, much easier to interpret at a glance.

What's nice is that modern cockpits, even for small GA plane, can easily have both. For example, my panel has an autopilot with Turn Coordinator/Slip Skid indicator, and my Attitude Indicator (AI), a Garmin G5, has a Turn Rate indicator and Slip Skip indicator.

The AI includes a 4 hour battery backup independent of the plane's battery and electrical system.

And in case the AI should die, a second G5 set up as a DG/HSI, with its own 4 hour battery backup, can be changed via menu selection to be the AI, so partial panel relying on the TC/Slip Skip, Altimeter, and Vertical Speed Indicator, and compass should never be an issue.

Doesn't mean they won't practiced any more tho.