If a headstone does not have "הלוי" or a pitcher symbol, could they still have been a לוי?

I’m a BT, my parents are not religious but are convinced that my paternal grandfather (who passed away when my dad was very young) was a levi and I’ve been living as such. I have done some digging over the years and found what I believe to be the headstone of my great-great grandfather in a small town Jewish cemetery in England from the 1920s. There’s no mention of הלוי or any special symbols.

If anyone’s knowledgeable about historic UK Judaism, is it possible they wouldn’t have mentioned it on a headstone even if he was a levi? I have no idea the religiosity but I imagine everyone back then was pretty observant?