tl;dr - if doctors don't denounce homeopaths, people will start going to "real" homeopaths and other alt-medicine people, and there is no practical limit to the lies and harm done by real homeopaths.

For a layperson there is no perceivable difference between a doctor and a homeopath.

That is so because doctors also offer homeopathy. If almost all doctors clearly denounced homeopathy, fewer people would choose to go to homeopaths, and these people would benefit from better treatment.

From the experience the homeopath might have more time to listen, while doctors often have a approach to treatment speed that reminds me of a fast food place.

This is a problem in its own right that should be solved by giving doctors incentives to listen to patients more. However, do you think that because doctors don't listen enough, homeopaths produce better treatment (i.e. better medical outcomes)?

they are already at a doctors place for treatment with placebos for trivial stuff, while actual dangerous conditions get check out from a competent person.

Do you have evidence that this is the result produced?

What if the reverse happens? Because the doctors endorse homeopathy, patients start going to homeopaths instead of doctors. Homeopaths are better at selling themselves, because unlike doctors they can lie ("homeopathy is not a placebo and will cure your disease!"). They are also better at listening, can create a nicer (non-clinical) reception atmosphere, they can get more word-of-mough networking benefits, etc.

Patients can't normally distinguish "trivial stuff" from dangerous conditions until it's too late - even doctors sometimes get this wrong. The next logical step is for people to let homeopaths treat all the trivial stuff, and go to ER when something really bad happens.

Personal story: my mother is a doctor (geriatrician). When I was a teenager I had seasonal allergies and she insisted on sending me for weekly acupuncture. During the hour-long sessions I had to listen to the ramblings of the acupuncturist. He told me (completely seriously) that, although he personally didn't have the skill, the people who taught him acupuncture in China could use it to cure my type 1 diabetes. He also once told me about someone who used various "alternative medicine" to eat only vine leaves for a year before dying.

When the acupuncture didn't help me, my mother said that was my own fault because "I deliberately disbelieved the power of acupuncture and so the placebo effect couldn't work on me".