Unfortunately ‘sourdough’ has come to mean two things: on the one hand it is a general term for all wild starters; on the other it is a particular term for actual sour bread, most particularly that made from a San Francisco starter which spoiled in the 19th century.
Like you, I find the latter really quite unpleasant (to the point that I really am surprised anyone willing eats San Franciscan sourdough). But wild starters can range from tasting just like ordinary commercial yeasts, to tasting similar but with a richer flavour, all the way to the funk that we both dislike.
I have never had a San Franciscan sourdough in Europe, so I doubt that this sourdough is the nasty (to my taste — many folks love it) kind, but maybe there is some subculture of inedible (again, to my taste) European breads. More likely, I suspect that this is just a good wild starter.
If you don't like sourdough, you don't need to make sourdough!
Use baker's yeast instead. That doesn't limit you to basic recipes -- there's a vast range of interesting stuff you can bake. You'll usually need to make a preferment with flour, water and yeast (a "poolish" or "biga") so the overall routine is very similar to sourdough.
I use a mix of bakers yeast and a wet (1:3) starter/sourdough. Just 2g yeast/kg flour makes a huge difference, even with 150 g starter.
My wet sourdough is 1 part flour to 3 parts water. As noted in a sibling comment, this favors the sour parts (lactic acid?), but compared to a dry starter, there are significant advantages the the starter/yeast combo:
1) Feeding the wet starter takes 10s: pour flour and water onto leftovers and stir quickly with a spoon. No sticky stuff to deal with.
2) The starter seems exceptionally stable, maybe because of the water layer: I only wash my starter jar every two or three months, and the 10g or so that I put back in the fridge after starting a dough, will last for weeks and consistently restart overnight when fed
3) Being able to independently adjust yeast-levels in a predictable way, means that I can easily play with sourness levels and adjust leaving times when I have to match the timings with other activities.
There is the downside, of course, that I need to keep bakers yeast in the house as well...
sourness is depending on quite a lot of factors - amount of water in starter, how long you proof, temperature etc.
I also prefer less sour breads and since I started using a stiff starter it's much better than more liquid ones. I still haven't found the perfect recipe yet but it is possible.
I think the main factor for this is how "mature" your starter is.
After feeding your starter it will expand, then collapse, then grow more sour.
I generally time it so that my bread isn't as sour.
If I am baking soon -- larger leftover starter and smaller feed.
if I am baking tomorrow -- tablespoon or so of starter with bigger feed to get my leaven.
As I understand it it's mostly the balance between lactic acid and acetic acid bacteria. There are different ways you can encourage the balance to shift one way or the other and they have different growth patterns under different conditions so timing your bake around it can work but may have other compromises. The yeasts are important too and are on yet another schedule.
You don't have to make bread. My current favorite thing is sourdough English muffins, but King Arthur has a whole list of recipes that use the sourdough discard. The sourdough pancakes are excellent, especially after you drown them in maple syrup.
You can. Twenty years ago "sourdough" didn't mean the finished bread was particularly sour. But people's expectations have changed, maybe because of the name. And it's easy to let the bread get very sour if that's the goal.
As others have indicated, this is indeed possible. A common technique that helps with this is to make a levain the night before. I use 1 tbsp of starter into 200g flour and 200g water. The levain develops overnight and becomes a "young" starter the next day. This produces a less tangy bread, compared to using a larger amount of full-strength mature starter.
