I know what you mean about cruise line hotels. I was in Copenhagen in 2019 for a cruise sailing Norwegian. They used one of the Scandics and I could not believe NCL's price.
I stayed at Hotel 27 but it has been sold and is now h27. It is on a side street but very centrally located, close to much. I had trouble finding it but I am bad with maps and cell phones. The hotel had a great breakfast. I do not know if they still do.
I bought The Copenhagen Card but regret doing so. I was unable to use it enough to make it worth the money. I suggest the hop on hop off bus. I did not use it but wish that I had.
I liked Tivoli Gardens which was an easy walk from the hotel and there is also Glyptoteket. I did not expect to love Glyptoteket as much as I did and I wish that i could post pictures.
There are incredible museums and castles. You will not be able to get to all that you would like to so do your research and make your decisions.
Restaurants and food is very very expensive. Research well. I did not. You will see many buffets that appear to be reasonably priced except that you are required to buy a drink and that makes the buffet much less reasonable. Some of the buffets do not advertise that you are required to buy a drink. I think that I asked for water not realizing that I had to pay the price of an alcoholic beverage anyway.
I did not do any organized day tours. There is very good public transportation and they offer various passes which includes going to specific attractions like castles and museums..
Check out Viator or Trip Advisor.