My best tip for always looking polished is to have treatments that save you time and effort if you can afford them. So for example, if I’m in a mad rush, I will always at the very least have eyebrows and lashes because I have my brows microbladed and a lash lift and tint. Gel nails are great as it lasts ages and doesn’t chip (if done properly) so you can forget about your hands for a fortnight. Tinted lip balm is quick to swipe on if you don’t have time to do lip liner etc. I also have a twisty Chanel thing in my bag, I know you said to look polished but for me if someone smells good they seem more put together. I never go to work without perfume.
Other than that I totally agree about greasy hair, it is my pet peeve and I make no apologies for judging people who turn up to work looking like an egg would fry on their head. I have waist length, thick, wavy hair that takes an age to wash/dry/style and needs doing every other day, so I truly know what a pain it can be, but there’s no excuse for uncleanliness tbh.
Oh and re heels - I personally do wear heels to work and feel better for it, but if you can’t walk in them, just don’t. Buy a nice flat or a small heel instead. Nothing worse than obviously not being able to walk in your shoes.