Living in the states growing up back east originally, smaller bars where mostly called taverns and they were usually packaged take out also. Now I live out in Colorado and they're either called a bar or a pub. But a pub would usually be referred to as an Irish style bar/restaurant here. I've never heard anybody say public house, even though the word pub is derived from that. When I saw the Irish word tábhairne, I knew it had to mean tavern, it's quite obvious.
It's pronounced "v" because that's the way many people pronounce it. There are also people who pronounce it with a "w" sound.
There's a lot of variability in how ábh is pronounced, and it doesn't follow strict dialect lines.
For example, listen to words like ábhar, [tábhachtach](https://www.teanglann.ie/en/fuaim/tábhachtach, sábháilte, *sclábhaí and you'll mainly hear a "w" from the Ulster speakers, but in tábhairne and gábh, the Ulster speaker has a distinct "v" pronunciation, whereas the Munster speaker pronounces tábhairne with a "w" sound.