Yes, it should still be part of new applications
I certainly still want it.
There are plenty of use cases where the alignment of the text is important and your are dealing with plain text files instead of word processor documents.
Two use cases come to mind immediately: tables and indentation. With a word processor you would use the table and/or paragraph style features to get the desired effect and not worry about using spaces or tabs to align your text in columns and rows or to convey hierarchy by using indentation.
For plain text files, the only way to achieve a table or indentation is to use a monospaced font and use spaces or tabs to align the text. Tabs are great if you only need a few tab positions, but how are you going to tell someone else's plain text editor where the tabs are supposed to be? So you need to rely on convention and hope that any and all collaborators use the same number of spaces for a tab, especially when their editors have such niceties as "translate tabs to spaces" and/or vice versa.
In other words: for plain text files the most reliable way to get consistent alignment for either tables or indentation is to use spaces.
Some Column Other Column Yet One More
Abcde 1234567 1
Abe 12345 3
Abcdef 123456 5
Acd 123 7
while cccc do
if a then
Keeping indentation isn't usually much of a problem provided you are using a proper IDE or good text editor (Notepad++). Trying to do the same with Notepad or some other plain text editor is tedious to say the least. Trying to keep a plain text table properly aligned is just plain tedious and cumbersome, even with a proper IDE or good text editor.
Having a replace mode available at least reduced the number of re-alignment adjustments you need to make. Admittedly, if you are not used to replace mode, you tend to forget to use it and thus use it even less. Keeping a table align with a minimum of re-alignment adjustments would require continual switching between insert and replace mode. Something only someone that needs to keep plain text tables aligned on a regular basis would be adept at.
I also often forget to use replace mode when it would have been a quicker way to achieve what I wanted to do. Even so, I still use replace mode at least a couple of times a week. Not having it available would annoy the heck out of me.
That said, most use cases for replace mode probably are limited to cases where fixed width fonts are used and alignment is/can be important. That means plain text editing of all kinds of source files: programming code, html, css, ..., ...
I guess that if you are building a word processor you might get away with dropping replace mode. But if there is even a remote chance that people would edit monospaced font tables with your word processor, they would be greatly helped if you didn't.