OK so hereās my take. Postman in my opinion is not a āfull fledgedā automation tool. It is a fantastic tool, however, for quick API testing, setting up test scenarios in collections and doing limited amount of automation. When it comes to source control, updating tests and Newman integration, the process is quite tedious and cumbersome. It sure can work, however you need Postman app to develop tests, you cannot really do it in the code, editing .postman_collection
JSON file unless youāre a total nerd 
Not to mention the huge debate on Postman forums about security and privacy. Allegedly dozens (hundreds?) of companies around the world are rethinking whether to ditch Postman completely and choose some other tool due to those concerns. Problem is, thereās not decent alternative to Postman, itās like switching from Windows OS, for Christ sake 
What I missed in most other comments is the fact that automation software should be as close as possible to production code. For example: your backend is in Java and UI is in React? Well, choose one of the two, whichever will be closer to your tests and/or whoās developers will more help in the automation effort. That way you can share snippets of code with developers, or even full features, components or libraries. Unless, of course, youāre Google and have a sea of SDETās at your disposal in dedicated teams, doing just automation 
Another fact to consider is: if youāre testing only the APIs, do you absolutely need a framework that is best utilised for UI testing? Perfect example is Cypress. Again, itās a fantastic tool but if you only need it for API tests and UI will never come, it might be too big of an effort for your projectās needs. That is of course just my personal opinion.
I am in a similar situation, I need to choose automation framework as best as I can. We have only APIs, zero UI. Our code is TypeScript/NestJS
and devs use Jest
for unit tests so naturally, I chose TypeScript with Jest
and supertest
. I might change my mind in near future, however, depending mostly on clientās needs. For example, I miss some out-of-the-box test runners and reporters like Selenium, Cypress or Playwright have, but Iām not there yet.
What Iām trying to say is (TLDR): choose the right tool for the job, not what you feel most comfortable with. I know itās a tough one but thatās the only proper way.
Last but not least, and forgive me for being blunt, I am a bit taken aback by the title itself - āPostman or Playwright for API testsā. Itās like giving us the answer together with the questions and thatās certainly not a good QA best practice. Thereās myriad of different tools. Yes, your options might be Postman or Playwright, but thatās a specific use case 