Not quite. Once you come close to the quest target you have to use your eagle to mark it. However, you still don't have to use the eagle and can still pinpoint the quest target yourself. An example? You need to free a man inside a fort, somewhere let's say east from the questgiver. Go east, find the fort, on a message "target nearby" don't use eagle, explore the fort completely and you'll spot a cage somewhere with a prisoner inside, free the prisoner, escort to safety, quest solved.
Note than in some cases the eagle can't help you. When you're hinted there is a cultist clue nearby (and you know that because someone told you or you've read it otherwise no hint on screen) the eagle can't possibly know where exactly to look for, the poor thing is as illiterate as dubs loving audience.
Thus you need to explore the area thoroughly yourself (basically reread the clue if you forgot what are you looking for, for example you need to buy a specific item from a blacksmith).
You don't have any icons anywhere unless you want to. Even if you're not using exploration mode, you can disable all icons on the world map. All. Yes, all.
If you mean questgiver icons (plumbobs) in live world, yellow or black, then yes, you can always see those above questgivers' head. But wait, I think even that can be disabled, lemme check.
EDIT1:
The game is updating, seems there is some new patch, wait some more pls.
EDIT2:
For god's sakes the game doesn't want to spawn me radiant questgiver. Wait, I'll try to force it somehow and post pics.
EDIT3:
Seems the patch did something, I can't find a single random radiant questgiver on streets. Sorry.
Anyway, pics.
1. All things enabled, as I've finished everything it's not cluttered much:
2. All things disabled - I didn't use this preset:
3. Why I didn't use "The Disabled" preset? Well because of the warning. There are 4 HUD presets (all, light, minimum and disabled), but you can enable/disable things one by one as you please: