I cook dinner without a recipe but every cake I bake is dense — what is actually different?

The difference is feedback. Cooking is a loop: you taste, you adjust, you correct right up to the plate. Baking is a single irreversible reaction where you assemble a structure, put it in a box, and find out 40 minutes later, so all your skill at correcting has nowhere to act. Everything you described is a structure error rather than a flavour one. Microwaved butter is partly melted, which means it will not hold the air you are trying to beat into it, and mixing until smooth is mixing past the point where the gluten stops helping and starts making it tough. Butter at cool room temperature, beat until visibly paler, and stop the moment the flour disappears.

For years I creamed butter and sugar for about 30 seconds because it looked mixed, and produced exactly the cake you are describing. Properly creamed is three to five minutes with a mixer, and it goes from yellow and grainy to almost white and fluffy, roughly doubling in volume. The first time I did it properly I thought I had broken something. That one change fixed more of my baking than every other tip combined.

Buy a scale, it costs about the same as two bad cakes. A cup of flour weighs anywhere from roughly 120g to 155g depending on whether you spoon it in or dig the cup into the bag and pack it, which is a swing of about a quarter, and a quarter more flour is exactly what a dense cake tastes like. I weighed mine ten times to check and the spread was real. Once you weigh, the same recipe gives the same result twice, and then you can actually learn from your mistakes.

Before you buy anything, put a thermometer in your oven. Mine ran 20 degrees cold on the dial, which does not matter at all for a roast because you just cook it until it is done, and matters enormously for a cake because the rise depends on setting the structure before the leavening runs out. Domestic ovens being wrong is the rule, not the exception, and it is the reason a cook who never has trouble with meat can be baffled by cakes.

Cooking is adjustable, baking is a formula. Weigh everything, control the temperature of the butter and the oven, stop mixing early.