The level of service you recall from Germany was common in the U.K. until a few years ago, or certainly ten years ago this was my experience in every practice area I lived.
GP practices are all private in the U.K., the visits are only paid for by the NHS. Precisely why GP wait times have gotten so bad I can’t say, but the real-terms funding cuts the NHS has experienced over the past decade likely contribute.
Of course, you can top up with private care in the U.K. also, and this seems to be becoming increasingly common.