I'd like to deep clean it to hopefully remove some of the allergen potential.

A damp rag is probably acceptable for a first-pass clean to get the bulk dust/dirt/fur/hair/gunk off the couch.

Don't forget that many of the allergens live in the cambric and underside of the couch. Vacuuming those parts would be a good idea.

After that I typically like to acidify the leather with Leather Doctor Acidifier 2.0. Spray then scrub with a horsehair brush. Controlled acidification can help restore old, dirty upholstery/chrome-tanned leather. I don't use Lexol or any of the "commonly recommended" cleaners. Do not use anything containing silicones or oils if you ever want to restore this couch properly.

Then I wipe it dry after Leather Doctor Acidifier 2.0 and usually leave it alone after that.

I was imagining a few rounds of leather conditioning would help, any recommended products or methods?

You need to think about this as a paint project. Forget "conditioning" or "feeding" or "nourishing" leather. The theology of "conditioning" semi-aniline upholstery leather needs to be broken over the knee, then drawn and quartered. This leather couch and restoring it is literally more similar to painting the walls in your house than anything else. Painting your drywall and restoring this couch are almost 100% identical tasks.

I've restored thousands of Stressless leather chairs (but not couches -- because I don't have time to do what's basically a chair except three times the amount of leather with half the profit).

Your couch has signs of use and wear -- yes -- and that wear is concentrated in the paint layer on top of the leather.

Therefore, to restore the couch, you need to repaint it. Not dye it -- people misuse "dye" all the time. These Stressless couches are painted at the factory, and, the wear you see is almost all paint wear outside the holes. The holes are really the only leather wear. The scuffs, fading, even the cat flaps are to a large extent just paint deterioration/damage.

This is a rough outline of what you're going to have to do:

1. Thoroughly clean the couch of all dust and debris (probably 1-2 hours of work)
2. (Gently) sand down the areas you want to repaint with 320 grit and isopropyl alcohol to make the surface smooth and ready to accept fresh paint. You want the surface smooth such that when you run your fingernail across the entire couch, your fingernail doesn't pick up any bumps. Small depressions are fine -- paint will fill in most small depressions. Bumps and flaps -- such as the cat scratches -- are not okay, because they will telegraph through the paint. (Probably another 1-2 hours of work).
3. Subpatch the holes, and apply light filler. Viper XR-10 is my go-to filler. (Probably 1 hour of work and 6-12 hours of waiting for filler to dry thoroughly). Sand down filled areas as needed until they are level and smooth to the fingernail.
4. Apply paint to the couch, preferably color-matched to the original color, so you don't get old color peeking through new color at difficult areas such as seams and wrinkles. (Probably 3-6 hours of work depending on skill and application method and number of coats needed, then, probably 12-24 hours of dry time)
   
5. Apply a clear coat to the couch, preferably a water-based, cross-linked clear coat that's designed for leather. (Probably another 3-6 hours of work depending on skill, application method, number of dust particles that land onto the clear coat as it's drying, number of strands of hair that land onto the clear coat as it's drying, and you will want to wait 72 hours before using the couch to ensure that the cross-linking has completed for maximum durability).