I'm embarrassed by those projects that failed where I failed to apply lessons learned in the past, but for the most part those are incidents.
The majority of the 'public failures' failed for new or unexpected reasons and I have absolutely no qualms about being open about it or analyzing them (in private or in public).
I'm far more suspicious of people that have never failed or that feel that don't want to talk about their failures. Pointing the finger of 'fail' at yourself and taking responsibility for mistakes made is a great way to improve your game and to increase the chances of eventual success.
As for dealing with this when promoting the 'next thing', I don't make any links between projects and I try to keep my person 'out' of the day to day running of any of the projects that I'm involved in, for the most part I'm the invisible man. So whatever failures there were if they might be problematic it does not carry over. But I'm actually not aware of any of that. Most projects, even the failed ones will leave you with the net positive goodwill that results from trying and trying hard.