Though one experience I will never forget going into a fancy bar that opened in my mum's small town and asking for a martini and the bartender looking confused at me and serving me a shot of pure unchilled vermouth in a near-empty tall glass...
The bar you visited was just ahead of its time.
Cocktail bars are a distinct thing, and have staff who actually know what they're doing, unfortunately they usually have exactly the same problems described in this article of being uncomfortably loud.
Bars in the city of of London, in the hours after work, are far from quiet.
A suburban or village pub may be different. You get to know which one in your local area suits your needs. e.g. I could point you to a local pub with craft beer and a quiet cosy corner, and suggest to avoid the one with the loud sports TV always on.
In the actual City area, with the tall buildings, the quiet kind will be nearly impossible to find at 6pm on a Friday.
There are absolutely quieter pubs available, but you need to know where to go.
Good traditiona oubsl, even the family friendly, are usually quiet.