Yes. Definitely. Without a doubt.
The problem with these old COBOL systems is that they have decades of patches one on top of another, and very little formal testing. These systems were made in a time long before "modern good practices" were established. They work because the business requirements are straightforward and change very little. And the things they do are relatively simple. The barrier to entry is extremely high. COBOL is not taught anymore, and even if you learn COBOL on your own in Linux, in real life it won't be a Linux OS. It'll probably be several layers of proprietary IBM VM emulation, with Linux running AS/400 running AIX. And on top of that, you have whatever customizations this particular user made. You're a slave of what someone that wasn't necessarily a "wizard" decided 40 years ago.
With a more "modern" language, COBOL can make use of modern "good practices", especially automated testing and such.
the "jump frameworks every couple of years to whatever is trendy" is out of place when you are mentioning Java and C#. Both are well-established languages and have been stable for literally decades now. Java and C# (actually .NET) people are not in the same game as JS developers.
The problem isn't the language, but all of the things that come around it. Using a modern language would, if anything, let you ditch the expensive IBM support contracts for mainframe hardware (and maybe switch to slightly less expensive support contracts for regular hardware)