Most international health insurance companies, definitely travel insurance companies will cover house visits at least for acute medical problems. Many of them will pay the extra cost even for vaccinations.
Ask any of the private clinics in Budapest whether they are able and willing to visit your baby with high fever, your husband with a paralyzing back pain or your wife with a migraine attack during the morning. Or what to do if your toddler fell against the table with his head. Consider the advantage of being able to rely on a doctor who checks your daughter/son in your arms in her/his room before deciding whether to send your little one for testing, maybe even a hospital stay?
Not to mention the other children you may have and have to drag with you to the clinic.
House visits are VIP service and available! If you have a good insurance they are available for free to you.