-bash: cannot execute binary file when opening an installer

The following looks to be the replacement for the Oxford New Headway Plus app:

https://apps.apple.com/us/app/oxford-learners-bookshelf/id617832312

New Headway Plus is now available as an e-book, with interactive features designed specifically for language learners. Available for iPad and tablets for Android via the Oxford Learner’s Bookshelf app.

What I see of the app posted around looks too old to work with Catalina, too.

The particular app here was replaced with new materials available from Oxford, so that’s what most businesses will want you to use; the most current version available from the folks at Oxford.

Catalina requires 64-bit apps. Older 32-bit apps will not work on macOS Catalina. The app developer has to rebuild the app for 64-bit, and possibly making some app source code modifications that can be required as part of that.

I see a number of different copies of what purport to be that OS X app posted around the ‘net, but who knows what those are, and what those contain. If someone at the business suggests loading random software acquired from the ‘net, you’ll want to check with the business IT, networking, and/or security folks about the wisdom of that.

The install icon likely isn’t opening because the app isn’t compatible. You can verify this by examining the app either directly with Get Info or related on the executable files, or with a tool such as Go64, and see if the executable files contain code built for 32-bit or 64-bit. But as stated above, anything listing OS X versions as old as this DVD is vanishingly unlikely to work with macOS Catalina.

If it says those are the minimum, then it is almost assuredly a 32-bit program that will not run on Catalina. 10.5 was 32-bit only. 10.6 was the first OS that could run both 32- and 64-bit programs.

Try

open /Volumes/new_headway_plus/setup-osx.app/Contents/MacOS/installbuilder

Is it a 64-bit program? If it is a 32-bit program, it won't run in Safari.

None of that was programming or codes.

However, as MrHoffman has pointed out, it likely won't run on Catalina no matter what you do.