Why is this even that complicated? It doesn't have to be an either / or question! I use The Trainline app a fair bit for short distance etickets (no booking free for on the day eticket purchases plus a friendly UI and the ability to show your Railcard directly from the app) and they show the eticket in the app, email me PDFs and I think have a button in the app to let me add it to the Google Pay wallet (I can't remember 100% for the last one on Android, but they defo do Apple Pay wallet on iPhones) which mean I can choose how I want to use the eticket. Isn't that the best option?
As for PDF's themselves - it is worth saying that they can be very useful if you are buying tickets for other people or to be able to have as a backup (PDF readers are more reliable than ToC apps in my experience, and having a PDF also potentially means it is easier to have the same ticket on a secondary device like a tablet or even a printed bit of paper incase your main device breaks or runs out of battery etc).