It's very good that you are addressing the error use case.
In this case you can drop the catchError cause you are basically passing the error forward which is the same result as not placing this operator.

Yes this is another approach you can take and use just the subject and not the observable.
don't forget to unsubscribe from the subject or you will have a memory leak.

You used both a subject and an observable created with fromEvent, why use both?
Why not use the operators directly on the observable?
Also you have a memory leak you create a subscription from the observable and the subject and you only unsubscribe from one of them.

Subject is a bit more fitting here, since you don't really need to initiate the subject with an initial value (the empty string is not really relevant here)
Also you don't need to access the last value the subject holds
also when subscribing you don't want an initial jump of the listeners.