It sounds incredibly expensive and not likely to give you any better insulation R-value vs. putting in fiberglass insulation--probably worse. You can fit R-13 or R-15 insulation in a 2x4 cavity. Even if I was wrong, I have to imagine that any cost savings over traditional insulation would be more than outweighed (no pun intended) by the additional cost of solid 2x4 walls.
The ideas in the article are mostly with finished materials, where you have the choice of using heavy or light materials--but as MMM stated, he didn't do all that *just* for the thermal mass--it was just a side benefit. Also, as I see it, thermal mass is really a complementary concept to insulation--they both slow the temperature change, but in different ways. I still don't think any "dual effect" of solid 2x4s would overcome the cost.
I have a similar issue by the way, very old house with plaster and lath most places. I would put in fiberglass in, at least under the attic floor if not the walls, but unfortunately I also have lots of knob and tube wiring, and as best I can tell it's not worth it to spend thousands of dollars to replace the wiring for the vastly smaller potential savings on energy costs. I've started with the more cost effective fixes like sealing doors and windows and outlets to reduce "leakage", as well as CFL/LED bulbs and the like.
I did start a project to staple some insulation to the attic roof where the attic is unfinished--it's a walk up attic with plaster walls and ceilings in 2/3 of the space--but ran into some ventilation issues because the stucco soffits have no vents, and I'm worried they'd sag and fall off if I cut holes in them--they've already been repaired once for sagging. Replacing the soffits with wood, repainting, etc. started to look like more cost than it would be worth. In the meantime I cheaped out and just laid some insulation over some of the attic floor, since we don't really use it except for storage (thermal mass!).