As @jonrsharpe explained, the problem is that FileBackend does not return proper Task objects. Rather, it returns raw objects with the properties of a Task, which do not have any of the methods. I wrongly assumed the Task objects were automatically constructed thanks to the type hints, but this was not the case.

The solution would be to construct a Task object from the retrieved items. For example, defining a factory method like this (the dates are being parsed using Date.parse() per as the MDN documentation):

public static tryFromRawObject(obj: {id?: number, name: string, status: string, creationTime: string, lastModificationTime: string}): Task {
    let parsedCreationTime = Date.parse(obj.creationTime);
    if (isNaN(parsedCreationTime)) {
        throw new TypeError(`Failed to construct Date object from value: creationTime=${obj.creationTime}`);
    }

    let parsedLastModificationTime = Date.parse(obj.lastModificationTime);
    if (isNaN(parsedLastModificationTime)) {
        throw new TypeError(`Failed to construct Date object from value: lastModificationTime=${obj.creationTime}`);
    }

    if (!Object.values(TaskStatus).includes(obj.status as TaskStatus)) {
        throw new TypeError(`Failed to construct TaskStatus object from value: status=${obj.status}`);
    }

    return new Task({
        id: obj.id,
        name: obj.name,
        status: obj.status as TaskStatus,
        creationTime: new Date(parsedCreationTime),
        lastModificationTime: new Date(parsedLastModificationTime),
    });
}

And then invoke it from the TaskRepository:

private async readStream(): Promise<Array<Task>>{
    const tasks: Array<{id?: number, name: string, status: string, creationTime: string, lastModificationTime: string}> = [];
    tasks.push(...await this.backend.json());
    return tasks.map(Task.tryFromRawObject);
}