David, that’s an interesting issue that I haven’t personally experienced, but having done a lot of flat-ground running training in the past, I believe you’re correct that speed work to improve your economy could be helpful. On hills, your legs just don’t have to turn over as quickly – or stride as far, it seems – to get your heart rate up.
My running training has revolved around Jack Daniels, who explored (and originally researched) a lot of the concepts contained in TFNA/TFUA, and Daniels’s plans always include several weeks of 200- to 400-meter repeats at about your 1-mile race pace with full recovery between reps. He refers to this pace as repeat or R pace.
If you have run a road race recently, you can use that to calculate that pace. If not, you can do it by feel – it’s basically just glorified striders, going fast but relaxed. It doesn’t correspond with an HR zone, because you don’t hold the pace long enough to get your heart rate that high. The idea is to purely stress your legs mechanically and to improve your efficiency at speed, hence the full recovery between reps. In my experience, it always felt like a light workout, and yet I would be a bit sore the next day if I hadn’t been training seriously.
Maybe give these a try once a week for four to six weeks. Start with 200m reps and bump them up to 400s after a couple weeks (or a mix of the two). Make sure your total distance spent at this pace isn’t more than 5% of your total weekly mileage: Even though these workouts may not feel that hard, they do introduce a new stressor to your training and will require some recovery. After a bout with these sessions thrown in, your legs should have an easier time with the turnover and power required to hit zone 3 on flat ground.