I self-identify as English as you suggest. I would support an English national anthem, though frankly that is at the bottom of my list of issues.
I reckon I identify as English first because I grew up in rural North Wales, I was consantly 'bullied' for being the new lad from england (even years after moving). I got physically bullied and spat at by 2 adult (40s-50s ish) strangers when walking to the local shop while wearing an England football shit, keeping in mind the world cup was on and I was not much older than 11.
The same corner shop was owned by an eccentric Englishman with a 'posh' English accent and his shop was vandalised on a number of occasions, he had to install shutters only for them to be spray painted with "Wales not britan".
I was once in a pub and a man was giving a pair of old English tourists some vile verbal abuse over a pint, calling them cunts and wishing them and all other English people dead. This went on for some time and I couldn't watch this poor old couple sit there and take that, so I told him in Welsh to shut up or leave. He replied to me in English "what? I don't speak Welsh" and attempted to blame that on the English tourists, until I pointed out where I was born.
All of that has imprinted on me as a young fella, I didn't see why school bullies had to pick of me and call me English scum. I didn't get why what football shirt I wore mattered enough to have spit coming my way.
Now I understand these things were more about cunts being cunts, rather than some deep seated anti-English sentiment. But it happend on such a routine that it pushed me to feel that being from England was a thing I should actually place importance in, because whether or not I do other people certainly do.
I also struggle with the idea that English nationalism is bad but other home nation nationalism is OK. And that is certainly the experience I've had of Welsh nationalism. I'm not suggesting to support a stronger sense of English identity or nationalism, I am suggesting my experience of Welsh nationalism is no better than the fella down the flat roof pub moaning about eastern Europeans. I think all nationalism of that sort is bad, but thats another story entitely!