To be more accurate, as I only glossed over your post the first time:
They are vector files not raster files and give a much better resolution in print, or on screen.
No they aren't. EPS is a container file. The E means encapsulated. An EPS can have vector images, raster images, fonts, PDFs and other file types jammed into one container. We used to send pages to a RIP that way all the time. And that was over 20 years ago. It's changed a lot since then.
Using an Adobe Photoshop file for instance to gain the transparent background needed is only a bodge.
Again, old thinking. Our main client used PSD files for all raster image placement in their flyers, catalogues and such for over 10 years before we retired. They hadn't touched an EPS since switching. Any vector images were placed into InDesign either as a PDF, or directly as an .ai file. Which works because for well over a decade, an Illustrator .ai file actually is a PDF.
EPS will b around for some time to come unless apps like Adobe Illustrator sort a different file format,
Again, an Illustrator file is just a PDF with a different file name extension so they open in Illustrator when you double click one on the desktop. There is zero reason or advantage in saving them out as an EPS.
but keeps the clean vector that Illustrator produces, WITH a preview icon…
And if you simply placed an .ai file into your page layout app, you'll still have a clean vector image and a desktop icon.
Really, truly. Stop using EPS.