Is fixed-scope, fixed-time really impossible to deliver in agile? Or what else use?

If you have fixed scope, and a fixed deadline, then the only thing you have left to play with is cost. You can throw more people at the problem (which doesn't really work), you can buy premade software, or you can sacrifice quality. ...Or you can change peoples' minds about the fixed scope or fixed deadline thing.

That's not an agile problem, that's a problem for all projects. Changing how you go about executing the project doesn't change the inherent constraints.

Yes, an agile approach could help you get the work done¹. At its core, scrum provides a way for a group of motivated individuals working together to deliver a product. Scrum provides a framework for breaking a larger body of work into smaller pieces (epics, stories, tasks) and then working on those smaller pieces.

Scrum also provides a framework for the team to constantly improve, which may also help get the work done. Finally, it makes the process visible, so that stakeholders can see how the project is progressing.

¹ An agile approach will help you organize and do the work, but it can't guarantee you'll finish on time or on budget. Agile is less focused on budgets and deadlines, and more on delivering the right product. That being said, even with a fixed scope and fixed deadlines, as a tool for organizing your team to do the work, it can definitely help.

Every project has to cope with the triple constraint of cost, time and scope. In your case time and scope are fixed. You say nothing about cost, but as you've inherited this project from your partner, I fear that there might be a fixed (or at least capped) cost as well.

But constraints don't kill project (although they can make them difficult). It's risks that kill projects and lead to failure:

• risk of quality: If you adopt a waterfall approach, you might find out at the end that there was a fundamental misunderstanding about the requirements or the customer's expectations. Cost and time needed to correct such errors can lead to an expensive total failure!
• tunnel effect: With waterfall, you'll enter a tunnel and you might find out only at the end of your design phase that the project was completely underestimated, and that you will need more time and more money to complete. As weeks will have passed, the customer might be very angry to hear bad news so late.
• self-fulfilling prophecy: with longer phases, there is a risk of the tasks taking all the planned time including margins in the schedule. For example people might take time to draft and redraft a perfect design document. But this precious time might lack in the end. Short time-boxed goals reduce such effects.

An agile approach is in my view the best approach to anticipate and master the biggest project risks. It allows to address uncertainty early, to get constant feedback about tangible progress, and to react as fast as possible.

So agile is not the problem, it's the solution.