I found some problems on Google Cloud managing paths like /{organization}/{project-id} with some tools (mainly BigQuery with automated transformations).
Because of that, one approach is to create a project without organization (with a billing account connected), but I am not sure about which possible future problems this will imply.
The only disadvantage I see is that we wouldn't use the GCP organization structure to manage IAM permissions.
If you have only one project, having an organisation or no changes very little (some features aren't available, such as organization policies - but of course you don't have an organization!), hierarchical firewall rules, or security command center).
You will have a real difference if you manage several projects: IAM is a use case, Cloud Logging sink and asset inventory.
However, it's safer, and cleaner, to solve your previous issue with organization/project that you have previously than creating a standalone project.