Dear Josef,
Greetings! Thank you for posting in Microsoft Community. We are happy to help you.
1.Regarding the first issue that SharePoint default landing page is in classic experience and you are not able to change it to modern, it seems that you have set a classic experience SharePoint site as default landing page.
May I know if my understanding of the scenario is right?
If yes, I'm afraid there is no build-in way to change a classic SharePoint site' home page to modern experience, the "Return to classic SharePoint" option doesn't exist on home page. Thanks for your understanding!
And it's expected that the document library, list, pages library and site contents are in modern experience when you access them via left side bar, they have option "Return to classic SharePoint" to return to classic experience.
Personally, I do understand your concern and I recommend sending your valuable feedback to SharePoint Feedback Community, it's specific platform to let the developers know your requirement and improve the product.
If you are willing, you may try to create a new modern SharePoint page and then set it as home page by going to Pages library, select the new page and click "Make homepage".
If my understanding of the scenario is different from yours, feel free to point it out, I will continue helping you.
2.Regarding the second issue that it doesn't work after you change SharePoint home page to new SharePoint site, generally, it might take up to 2 hours if you switch your home site to a different site.
If you have already waited for 2 hours but it still doesn't change to the new SharePoint site, we do understand the inconvenience caused and apologize for it. We're afraid it may need to check the situation from back end and fix it from back end. The best way to troubleshoot the situation further is to report it to the related team. They have higher permission and more resource than us, this would be the most efficient way in handling this case for you.
Microsoft 365 administrator has right permission to contact related team, therefore, we sincerely recommend you let your Microsoft 365 administrator follow steps in this article, raise a support ticket and contact Microsoft related team.
For reference: How do I find my Microsoft 365 admin? - Microsoft Support
Thanks for your cooperation and understanding! We hope you have a nice day and stay safe.
Sincerely,
Tina | Microsoft Community Moderator