Policymaking and public health guidance is complex. There are details of the population's behavior that can't be fully controlled, but can be influenced, and maintaining credibility is crucially important here. Burning public trust with lies (or worse, basic inability to think critically under conditions of uncertainty) early on in the pandemic was a lot more damaging than the incidental effect of a cloth-mask recommendation on early N95 supply would have been.
[1] Worse than and more plausible than lies is simply incompetence. The medical industry is famously bad at critical thinking: I've known people who came out of (top-ranked) med school _less_ able to reason under uncertainty and deal with statistics, because of the culture of oversimplification they're indoctrinated with. I have no opinion on whether this is wise for the field overall, but it's an especially poor fit for a fast-moving, low-information environment like the early days of a pandemic.
Fauci should be tried at Hague.
If the virus is in microdroplets in your breath, something in front of your face when you breathe out is going to slow some of those droplets down, making them less likely to be inhaled by others nearby. This is just physics. If it can fog your glasses its helping. If it makes you smell your own breath its helping.
A scarf is better than nothing, just like a homemade mask is better than a scarf, just like a purpose built mask is better than a homemade one.
"This is just physics".
No this is just your intuition. Scarfs don't protect you or others. Neither do gaiters. This has been tested. They may work mildly for UV protection... But there you can be fooled as well.
They just don't work. N95s provide a level of protection but they are not full proof. The further away you move from N95 the more useless they get. Cotton etc. don't protect. They are overcome by physics.
The reason Fauci (and others) said not to wear masks early on is because there was a shortage and they wanted to be sure medical professionals could have access to masks.
Medical professionals have known for a century that masks help prevent the spread of disease - particularly an airborne or aerosolized.