As for black holes being "infinitely small", you need to do some further research. No part of a black hole is infinite anything. The schwarzschild radius is the size of an object where its matter's physical properties exceed the laws of the known universe. Simply put, you can't have two objects occupying the same space. So if the overall size of the object is smaller then the amount of space it needs, it becomes a black hole. If you had two protons and tried to squeeze them into a space smaller than two protons, you'd have a very short lived black hole. If the Earth could be converted into a black hole, it would be the size of a pea. This isn't "infinitely small".
As for the idea that indirect observation isn't enough, you've never seen the wind, electricity, gravity, etc. but believe in them nonetheless.
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By changing a few words, this question could easily be in Religion & Spirituality.
Despite its invisible interior, the presence of a black hole can be inferred through its interaction with other matter and with electromagnetic radiation such as light. Matter falling onto a black hole can form an accretion disk heated by friction, forming some of the brightest objects in the universe. If there are other stars orbiting a black hole, their orbit can be used to determine its mass and location. Such observations can be used to exclude possible alternatives (such as neutron stars). In this way, astronomers have identified numerous stellar black hole candidates in binary systems, and established that the core of the Milky Way contains a supermassive black hole of about 4.3 million solar masses.