OK. So, how should I cover it?
I never cover a rising dough with gladwrap although I know it's a common solution; I place a cotton kitchen towel loosely on top of the bowl.
The temperature in my kitchen varies but that has never had a great impact in baking. I tend to keep the window closed (= draft free) and the bowl on top of the warming oven as you do - at least partly/for a while.
The yeast - whether fresh or dried - has never died of warmth in my experience although I warm the liquid by gut feeling - not with a thermometer - before adding yeast. Fresh yeast = body temp liquid, dry yeast/active yeast = slightly warmer but not hot liquid.
There might be site-, yeast brand-, flour- or handling-specific reasons for
Hungry Man's recipe not to work out in your kitchen. You could try my
Christmas Flatbread to experiment with working with dry yeast if you feel industrious. In that recipe, the dry yeast is mixed with just room temperature ingredients without warming. Soft flatbread forms after hours of rising. Thus, much more time is needed compared to
Hungry Man's recipe. The Christmas flatbread recipe can be converted in to a savory cheese flatbread by reducing the amount of syrup, changing the blue or feta cheese into mozzarella, leaving some spices, seeds and herbs out and adding garlic and scallions instead. The dough is completely different: soft and sticky and has to be flattened with floury fingers - but at least the end result is soft.
Dry yeast/active yeast works best if you let it wake up and form bubbles for ~15 minutes by adding all of the liquid, (some sugar/syrup if included in the recipe) and half of the flour - or by following brand/active yeast specific instructions (with lesser ingredients mixed with the yeast for a start). Subsequently, add the rest of the ingredients and let the dough rise as a solid clump after kneading (if kneading is necessary/included in the recipe; the Christmas flatbread is left unkneaded) until the dough has doubled or tripled.
If gladwrap is used during rising, the wrap is supposed to be set on top of a large bowl, not in straight contact with the dough. Dough is mostly wrapped in plastic only if it's supposed to cool/set in the fridge. For a "normal" yeast bread: when the dough has doubled/tripled, remove excess bubbles by kneading lightly and form the bread/flatbread and let it rise for the second time for ~15-20 min in a lukewarm and draft free place on top of a baking tray with a light towel on top before baking.