Ah yes, the play on the word assumption. Yes, science contains many assumptions, what is relative is the difference in (your) layman's definitions of assumption and that when applied to scientific research. In scientific research assumptions are based on past experience and testable criteria. Lets look at the radiometric dating method which was used, uranium/lead (U/Pb). What is assumed?
1. The decay rate has remained the same. Why is that assumed? It is assumed because it is a measurable rate that follows the laws of physics and is directly observable and never found to be variable. Tests have also been made exposing isotopes to conditions such as heating, cooling, pressure, chemical reactions, etc., exceeding those that would be encountered in nature. Rates remain the same. Isotopes emitted by gamma rays from supernova millions of light years distant have had their decay rates measured and they are the same as observed on earth today. Thus, decay rates are not an unfounded assumption. They are assumed to remain the same because of the observed data.
2. That there is no contamination. Actually that is never assumed. There are specific tests and criteria for detecting and quantifying any contamination (excess daughter isotope). Furthermore, there are methods of recent development where contamination is irrelevant, Atomic Trap Trace Analysis (ATTA).