I take my animals in for all their shots. I do the puppy/kitten series followed by a one-year booster and then do blood titers to test their immunity after that. Most dogs have immunity for about 3-5 years with the basic shots (that's after doing a puppy series followed by a one year booster). I just had to booster my 6 year old dog, after she tested very low immunity to parvo, but good immunity to distemper. Almost all vaccines are now every 3 years (including rabies), as studies have shown that provides adequate immunity. Some aren't, like the Purevax rabies vaccine is yearly, but it is non-adjuvanted and causes less reaction in the animal (so safer, but shorter immunity). I don't like to vaccinate any more than is necessary, and I only vaccinate if the disease is often deadly or hard to treat (I don't vaccinate for kennel cough, although in rare cases the disease can cause deadly pneumonia).
That being said, I'd recommend everyone get their animals vaccinated for rabies, at whatever interval your state requires. The reason it must be done by a veterinarian is to assure compliance and that it is done correctly. Every year 50,000 people die of rabies, just not in this country. And that is because of the vaccinations that we give our animals and that people know to report bites to their doctor. Wild animals carry rabies, especially bats (that can expose even indoor only animals). Some states can immediately euthanize an animal if it bites (for ANY reason) and doesn't have proof of rabies vaccination (the ONLY test for rabies is to send the brain to be tested), some only require quarantine (but it depends on type of animal). So if you are going to forego your rabies vaccine, check your state law first.
As far as whether you should do regular visits, I think of it this way... would an early diagnosis change what you do? Would you spend $2000 fixing a leg? If you found out they had heart disease would you treat it? Would you treat cancer? Would you spend $300-$500 on getting their teeth cleaned? If the answer is No, then regular visits can be skipped. If the answer is yes, then you should be getting regular checkups. My one dog came up very mildly lame, turns out he tore his cruciate ligament (ACL). If you don't fix it in dogs, they'll usually have crippling arthritis in a few years. It's something I plan on fixing (I don't want to have to euthanize my dog at 5 years old for lameness), but if you wouldn't then you can skip those visits. The one thing I think you should get them seen for is pain, as its unfair to leave a pet suffering when it can easily be stopped. And with skipping regular visits you should educate yourself on common signs of illness and important animal medical knowledge (like suddenly having potty accidents often means there is a urinary tract infection, fat cats can die if they don't eat for 3 days (fatty liver disease), changes in water consumption, food, sleep habits etc can all be signs of disease, and that sort of thing).
There are options for euthanasia at home, and many vets can do it at your house. For the exact reason that no one wants an animals last minutes to be stressful. The other option would be getting enough sedatives for at home and then taking them to the vet when they are pretty much unconscious. It's a tough enough time, without having to stress out your pet too.