To see, change or delete Script properties, the most reliable way to do that is to write some code that uses Properties Service.
For example, you can log Script Property values as follows:
function getAllScriptProps() {
var v = PropertiesService.getScriptProperties().getProperties();
Logger.log(v)
}
You can continue to use the legacy editor to view and manage Script Properties, but the UI has always had bugs in it.
It will display Script Properties, but it doesn't always update or delete properties correctly.
Script Properties are part of Properties Service, which is separate from the Apps Script API. Properties can be tied to a user, a document, or the Apps Script project file. Even though the Apps Script API is for accessing Apps Script files, it has no way to list or update Script Properties.
The capabilities of the new IDE seem to be closely related to what the Apps Script API can access.
If they do decide to allow the new IDE permission to access your Script Properties, and add a UI to manage them, it would probably be announced in the release notes.
Apps Script Release Notes
I'm guessing that the legacy editor will eventually be deprecated, and if a new UI to manage Script Properties does not get implemented, then you'll need to use code.