For primary training, I don't think it matters much although I prefer as simple an airplane as possible so the student is looking outside and really developing their stick and rudder skills.
I get so frustrated teaching the Cirrus because folks just can't take their eyes off those big, fancy displays. It's a problem with low-time and high-time pilots alike, they just spend too much head down time in the airplane.
For instrument and advanced training, I think it depends on what the student's goals are.
If they're heading straight to an RJ there's an advantage to be had with more exposure to glass, flight directors, autopilots, MFDs, etc... and putting all the systems to best use.
We have some foreign students sponsored by an airline coming soon. They'll never see a steam gauge professionally as their next job is going to be right seat in an Airbus.
I'm as old school as the next guy but I can't really see any reason not to get them looking at glass and deploying automation as soon as possible.
Likewise, if the student has plans to buy, or has already bought, a TAA (glass Cirrus, Cessna, etc...) and is going to be flying that airplane, by all means glass and automation should be stressed as early as possible.
Conversely, for your average GA pilot who may never see a glass cockpit it would be silly for them to train in one then have to transition back to steam gauges.
It's the old "train like you'll fly, fly like you've trained" adage.
The key is to ensure that in their training the glass and automation available to them is not a crutch, but a tool to allow a safer, more efficient flight.
Glass is just so much better and ultimately simpler to use than the old steam gauges, VOR to VOR systems of the past.
But the key is that the student needs a complete understanding of the system. 90% of instrument flying is mental and that's the biggest challenge I've found in training: Getting pilots used to thinking like an instrument pilot.
What instruments they look at aren't as big a factor as what they're thinking about.
If they have a good understanding of the system they should be able to jump back and forth without too much difficulty. I know this because I fly both glass and steam gauges (usually in the same day) and it's just not a big deal.
I love automation because it frees the pilot up to spend more time managing the flight, reduces stress and workload and ultimately should result in a safer flight.
I'd much prefer to fly a glass Cirrus IFR than a steam-gauge Skyhawk without a GPS or autopilot because it's easier, but I'm just as comfortable in both.
All that being said, the "old school" skills are still critical and when the screens go dark they need to know how to fly an approach with a few, poorly located round gauges.
And I agree: Knowing how to fly a DME arc properly or a raw data NDB approach is still a key skill that folks have to know.
As for background, I'm a full time CFI/CFII and Cirrus instructor with 1,100 hours or so. And I learned to fly out of a grass strip in an old 172 and did my instrument rating looking at steam gauges.
Good luck! I'm sure you'll get a variety of opinions.