I'm glad that you were able to resolve your issue and thank you for posting your solution so that others experiencing the same thing can easily reference this! Since the Microsoft Q&A community has a policy that "The question author cannot accept their own answer. They can only accept answers by others ", I'll repost your solution in case you'd like to "Accept " the answer.
Issue: OpenAI Project - 'Add your data' giving the following error:
'Invalid AzureCognitiveSearch configuration detected: No searchable fields in index'
Solution: Removing the existing index and creating it again, solved the problem.
If you have any other questions or are still running into more issues, please let me know.
Thank you again for your time and patience throughout this issue.
Please remember to "Accept Answer" if any answer/reply helped, so that others in the community facing similar issues can easily find the solution.