After PhD, no support from PhD supervisor
Let me answer the following question out of many:


  What if their supervisor says "you are not an independent researcher, so I don't think you could really sell yourself as an independent researcher?"


Based on the evidence you gave, your supervisor is right. You typically need 3 to 5 recommendation letters and some evidence for being able to acquire funding to get to the next level. If you get no support from your supervisor, these 3 to 5 letters have to come from someone else, and you gave no evidence from whom. As soon as you get a "yes" that these letters can be written, you can get them uploaded to a dossier service and reuse them for your next 100+ applications, but, the way things are described by you, you don't have 3 to 5 "yes", so, you cannot declare yourself independent enough. In short, you are not independent enough from the goodwill of your supervisor, and, since you probably don't have this goodwill, you might probably have more chances for positions that do not require this goodwill, i.e. (unless your internationally shine in your area) no assistant professorhsip positions this time, perhaps even not grants or scholarships.  Maybe later, maybe not at all, but not now. 

Otherwise you might simply lose months on writing 100+ applications and get zero outcome. If you sell yourself as independent while you are not, you simply spend time and money on letting others know how immature you currently are.

I would love it if you prove me wrong. I'm sorry for being that negative - I'm speaking from experience.