You can. I find the trainline app is best - it charges a small fee. Paying the fee is value for money compared to the awful ticket machines in Huddersfield
I'd never pay a fee to use Trainline!
If you buy on the day using the app, there is no fee, so if I can't get the ticket I want on Trainsplit, I will use Trainline on the day, if that will sell it.
Trainsplit never charge a booking fee, might save you money, and you can select your seat on any train that allows reservations, so I can't think of any reason to use the Trainline app over Trainsplit, unless you literally have money to burn.
I definitely do like having tickets on my phone. The only reason not to, would be if I was unable to have a backup.
A backup could be any other electronic device or a paper copy.
Another good reason to get e tickets is when splitting you get one PDF with all tickets on it; it makes it much easier to show the tickets and helps with Delay Repay claims (just attach the existing PDF).
The only caveat is to avoid apps that issue 'm' tickets as these are specific to a phone and have to be activated. They need to be avoided all costs. Proper 'e' tickets are absolutely fine.