Patchy lawn with moss and unknown grass type – advice before spring renovation

After following NZLA on Facebook for a couple of months, I’ve decided to join the group and start my journey

Past history of lawn care;
Past couple of years, annual spray of triclopyr based chemical for broadleaf weed control
Last year I purchased a scarifier (in November) but thinking I might have slightly damaged lawn (maybe to late in season), as yellow/brown patched in lawn have been a regular sight since the regrowth came through.
And application of a lawn thickener

Current condition of lawn
Patchy yellow/brown sections
Grass species unknown (my untrained eye assumes a mixture of fescues and rye)
Shaded areas prone to moss
From time to time, mushrooms and fungi are present

My aim:
The plan is to overseed with fine fescue in the early spring (will follow the NZLA Reno guide when I get to this stage), after running the scarifier over it again.

Between now and then I’m considering started to introduce winter care treatments (charger, nurture, Humic+, iron+, restore, and liquid starter)

Would I be on the right track, with my current plan, or bark up the wrong tree?

I’m excited to start my journey, and look forward to posting my results through my journal