I would have to say no to that one. If you look at how many sites that are ranked on the web most are on shared hosting setup with the same default name server the host gives to everyone on that server.
May you use private name servers or change your current nameservers, it is won't really affect any search engine rankings or “SEO Rankings”. It's important to take care that if you change nameservers it shouldn't be a fake one. Your hosting provider plays an important role in your SEO rankings. If there are any kind of changes in your SEO ranking, then you can be rest assured that there are many other reasons behind that.
I am using cloudflare name server, with that NS, i believe million website share the same name server. I personally don't think it affect your SEO. Yet, I do see people discussing unique IP for a website is good for SEO.
When it comes to check backlinks, competitor position etc I always prefer Ahrefs as it is easier for me to use.Someone have stated that the features are buggy and not relevant but I have not experienced this yet. When it comes to performance, analyzing both site functionality , features, services as a whole then also I will still stick with aherfs, it never let me down and proovided the data which I need. Finally, it's depend on you what you like is your decision.