Holmes in the original stories is most assuredly not courteous, empathic and kind as a rule - at times he is all of those things, re selected people and situations, but he is frequently none of them and it's not always towards people who deserve it, indeed his treatment of Watson is downright shoddy.
Further I think at this point we must accept Conan Doyle was not hugely consistent in his portrayal of Holmes and at this point, the character has been used by so many other writers across so many mediums that "Sherlock Holmes" has become a composite concept, almost a palimpsest, and his level of brusqueness, contempt and decency is highly variable. As is the Watson figure's capability and intelligence. So I think he works fine unless you're a purist, in which case most Sherlock Holmes based media is out.
As for "far from ideal for any doctor", obviously but have you met doctors? I would say a full 20% of doctors (GPs, specialists, etc) I have met have personalities and behaviours very unideal for their profession, whether they're rude/contemptuous, sexist, racist, bullies, incapable of listening, totally tactless, dismissive of serious issues, or all of the above and more. I would say fewer doctors have the mein of a saint than a jerk (though the saints do exist!). Obviously House is towards the worse end, but he's not as badly behaved as you can be and still have a job as a doctor, at least on the earlier seasons. The later ones push credulity a tad. He is at least typically interested in solving problems which makes him better than some percentage of doctors.
I have indeed met hundreds of doctors, being one myself, but have not met them often as a patient (half a dozen times, maybe) and have sat in on hundreds of GP consultations in my training and afterwards. Most doctors do their best in the short time they have with a patient and try not to be any of the things you describe, but sometimes fail.
(Full disclosure - I’m U.K. trained and have been a doctor for 23 years and a GP for 17 years. I’ve worked as a GP in England, NZ, and Canada.)
UK GPs especially have had dozens of hours of training in building rapport, non-confrontation, cultural communication, and other elements of consultation and communication - more so than most other specialities - and yes, it doesn’t always take or work, and especially these days doctors are often very stressed and pressed for time.
Being a patient is often awful enough without the doctor making it worse - you can feel helpless, put on the spot, unable to articulate what you exactly want or need, unable to deal with a system that is entirely non-transparent, and of course you’re in physical and mental pain, and there are usually many other stressors such as being at risk of losing your job because of taking time off work. A well trained doctor should know all that and at least know what the basic bounds of professional behaviour are. I’m very sorry if you’ve dealt with many doctors who haven’t appeared to understand this.
House, from the dozen episodes or so I’ve watched, is quite a bad doctor in a number of ways that frankly offend me professionally, even within the bounds of dramatic license and narrative TV tropes. He’s rude to everyone, including admin staff (which is close to unpardonable* in actual medical circles), and he frequently punches down with patients who know less than him (which is most people) - he humiliates and mocks the people in his clinic regularly for cheap laughs. In doing so, he often fails to elicit actual useful information or build a rapport which means they might actually comply with his treatment plan (which of course is handed down from on high and not explained or mutually agreed, oh no, why would House care what a mere patient thinks?). He’s often nicer and more empathic to the patient of the week - presumably because he spends more time with them - but he still often regards them as a puzzle rather than a person**.
*Seriously, a doctor who is regularly mean to receptionists or secretaries is an effing idiot and can expect to never see their letters sent on time.
**As for doctors being interested in solving problems, they’re always interested in that - that’s what they’re there for - but the question is whether they’re interested in finding out and helping with what the patient’s problem is rather than identifying and solving the clinical puzzle. It’s certainly been hammered into me in my training that my job is to do the former.