Yes, free T3 levels are a major determinant of metabolic rates and hypothyroidism leads to fat gain. Note that we should refer to functional fT3 levels, acknowledging that fT3 and rT3 determine the functional fT3 level.
Low functional fT3 levels affect many systems in the body, lots of things go wrong, perhaps in subtle confounding ways.
Restoring normal functional fT3 levels will increase body temp and heat loss will increase and metabolic rate will need to meet those demand.
"functional fT3" is something that I came up with as part of my understanding how rT3 changes everything. Note that the term "functional fT3" does not currently exist in Google search results in a thyroid or health context.
Other things control metabolic rate, steroid hormones [including vit-D25], GH, cell wall permeability, CoQ10 levels/deficits, nutrients/diet, activity, mood, cortisol levels, stress, insulin resistance, comorbidities, drugs [Rx and OTC], gut flora.