Here's the bad news. After a certain year (think it was around 2018? I am not sure), Apple changed how the apps are installed/downloaded onto/into Apple devices. It used to work like any software; when you clicked on an app to install it, the actual app software was downloaded onto your computer, phone or ipad. After that year however, clicking on and installing any new app has NOT downloaded or installed the app itself. It merely installs the software portion needed to RUN the application. So, any app you've installed since then has NO BACKUP of the app itself. It's not possible as the app was never downloaded in the first place. All you get is permission to use the app, essentially, and a backup of the files you create while using it. So if and when that app vanishes from the App Store - poof - it's a terrible and permanent loss.
The situation got even tougher then. A couple of years ago, Apple removed iTunes for app installations altogether. If a person knew about this change beforehand, they might have tried to save the actual app installation files (.ipa files) onto their computer, and hope that if the apps disappear from the App store in the future, they could somehow re-install them. First though, a person had to be able to figure out how to FIND those files, as they are hidden, as well as figure out where to save them so that they won't be auto-deleted by iTunes or the lack of iTunes, and that can be located again by a connected phone or iPad. All of that is a bit of a techy process. Now though, there is no iTunes to facilitate re-installing these apps. I am not techy enough at all to know if it apps can be reinstalled without iTunes... I suspect not.
But here's the good news. Since you have an old computer that still has an old NOT-UPDATED iTunes (from around 3 years ago at least, right?), and you have apps showing up on the computer's iTunes Purchased list when the phone/ipad is not connected - yay! You have full versions of those applications living on your computer's hard-drive. To my knowledge, you must have iTunes still working in order to reinstall that app again on your phone, which you also have.
Sorry, there are still more hurdles to clear though. Every time Apple updates your iphone/ipad with a new OS, many aspects are changed. This sometimes means mobile apps have to be designed/developed in a specific way either to function on newer Apple devices or because Apple has their own protocol to follow. In turn, developers of older apps must update their code in them. Lots don't, for any of many reasons (too busy, app doesn't make money, too much work, too difficult of a coding level, and/or they just don't care about it anymore.) So even IF you reinstall an old app, there's a chance it won't open, will be buggy, or will crash. The older the app, the higher the likelihood.
So, if you are not too discouraged, and are willing to dig deep into learning to reinstall an old app or can find an experienced tech-type person (who understands and knows advanced procedures beyond the standard ones found in Apple's Help files), there's still some hope. No guarantees though, sadly. I personally have hundreds of wonderful apps that have been removed from the App store, and will one day try to tackle this challenge on my own older Mac old iTunes backup too. Needless to say, I am not looking forward to it.
In the case of your phone not "speaking the same language" as your old iMac, you'd probably need to search the internet for a copy of that specific software your phone is asking for if Apple no longer has it available for download, but that is illegal in Apple world. Anytime you do, you do so against their advice, which voids any future device/computer repairs at all after that.)
Assuming you cannot find and install your missing "iphone to old computer translator", you'd have to locate the actual .ipa file for the app, and copy it into your Local Files folder on your phone (via Dropbox or and email or something.) - but I will not recommend it because Apple's online posting Rules do not allow this. You might then wonder if you could find a computer tech who knows how to get that file into a location where you could actually open and use the app. I am not sure if that is possible but again it is very very against Apple's online posting rules to suggest so I do not suggest that. As a last resort, your only other option may be to find and use an older phone or ipad that has not been updated past ios 12 to install that app there (as per normal Apple procedures for that device). Note: apps that are much older than 2016, won't install on any phoneor ipad thqt is has been updated pas ios 8..
I hope you find a solution!
