Welcome to the community!
I personally have both the Amazon Prime Rewards Card (Chase) and the Amazon Store Card (Synchrony)
I have never had any issues with the Amazon Prime Rewards Card and I have had it for a couple years now.
I will note that I have never shopped at Whole Foods which gets you a 5% cashback if you are a prime member, but I have taken advantage of the 3% back on Amazon purchases, the 2% back on restaurants and gas and all other purchases which give you 1%.
Using it strategically I have been able to get some decent cashback in the form of Amazon credit.
As to the fact that Chase tends to mess up people's credit, I will note that I applied for this card several times and got denied before eventually getting accepted.
The simple fact is that Chase has strict guidelines for credit score and income before they will even consider approving you and their 5/24 rule (link to an article by The Points Guy with more info) keeps a lot of people from getting approved.
Besides the several credit pulls over the course of a couple years trying to get the card, I am not aware of any other negative affect it has had on my credit.
Since getting the card, I have been able to increase my available credit several times which in turn has actually helped my credit score.
When it comes to the Amazon Rewards Visa Signature card, I would definitely recommend doing your research to make sure you meet all the requirements to apply, and if you do, I definitely recommend it as a great card.
Of course, the initial credit pull when applying for the card may affect your credit score slightly but as long as you don't max it out and keep the balance low, it should actually help your score in the long run.
Hope this is helpful.