Is the SPFx client web part permanent? Is this a good approach to develop websites for publishing in SharePoint Online (Office 365)?

You should check out the add-ins hosted by SharePoint providers for your case. This allows you to deploy your add-in as an Azure web application, create a site-to-site VPN with your local data center (if there is no express route in your area), and SQL Server on-premises data in your add use-in by hybrid connection.

According to Microsoft, the SharePoint Framework (SPFx) is a page and web part model that provides full support for client-side SharePoint development, easy integration with SharePoint data, and support for open source tools. With the SharePoint Framework, you can use modern web technologies and tools in your preferred development environment to create productive experiences and apps that are responsive and mobile from day one. The SharePoint Framework works for both SharePoint Online and local applications (SharePoint 2016 Feature Pack 2 and SharePoint 2019).

With SPFx, you may be able to build your application's user interface and use SharePoint data. However, you also need to think about connecting to your local SQL Server database.

Otherwise, you may need to consider making your SQL Server data available using a WCF service and making it available online in SharePoint via BCS. This has a limited scope for working with your data compared to provider-hosted add-ins.