How can I, nicely, get my dad (90) to do what he is capable of doing?

No, he is being "Dad". Dad gives orders, not takes them. You have to convince him that all the good ideas are his. Lavish praise. Reward with food. Feign incompetency, say "You are so much better at this than I am". By the way, I've been married 44 years and this works on husbands too.

Sometimes "won't" is "can't" with pride wrapped around it.
Maybe he could use shirts that look like button ups, but are really closed by velcro.
It's not unheard of for elders to give up something they can do simply out of fear, but I hear of this more with walking again after a fall.

Try "the doctor said you need to do it so your hands don't freeze up". "The doctor said...." was pretty useful with my mom.

My girl-friend was hospitalized so I drove over to her house and made dinner. Her husband stayed in his man cave until precisely 4:30 pm each evening, then toddled out to the kitchen, sat in his chair at the table and smiled.
He fully expected the meal arranged on a plate, the coffee fetched hot and the condiments within reach. They are married 50+ years.

Men do need special coddling sometimes, at any age.

He may be feeling a bit insecure about changes in his life and wants you to help anchor you to him, to your relationship, and just to feel that he's not facing old age alone.

New age psychology notwithstanding, maybe he really does have trouble buttoning his shirts. My father does, and tries to find shirts with dot snaps as often as he can. His accident with a radial arm saw a few decades ago permanently altered dexterity in one of his hands.

You can also suggest that you want him, and know he would want to as well, be as strong and dexterous as he can as he ages, and continuing to do these basic things will help that.

I never knew how clueless [or spoiled] my Dad [94] was until my Mom had recently passed. Dad told me he can manage on his own, but that first day being on his own I got a call at 8pm that he was hungry..... oh good grief, he can answer most of the questions on "Jeopardy" or "Who Wants to be a Millionaire", but can't make himself a sandwich or a bowl of cereal :P

Yep, my Mom had spoiled him. Oh how I wished she would have taught him how to vacuum, how to dust, how to do things in the kitchen, how to run the washer/dryer, and the 101 other things she did around the house. But she was of that era where it was "her job" to do everything around the house and spoil Dad. Wait a minute, Dad got to retire, but Mom didn't.

My advice, don't enable Dad.... let him learn on his own how to do things. I also got my Dad one of those shirts with the velcro buttons but I have yet to see him wear it. Next month my Dad will be doing breakfast and lunch on his own every other day, to see how well he manages, I am keeping my fingers crossed !!