Choose the measure and record it before the change. Nielsen Norman Group's guide to benchmarking describes comparing an experience's metrics with a meaningful reference: an earlier version, a competitor, an industry standard or a goal the team sets. For a change to your own site, the earlier version is the reference you control, and parts of it are hard to recover later if nobody captured them, such as observations of people attempting a task on the old page.
A before-and-after difference still needs careful reading. Other things change between the two readings, including traffic sources, the season and whatever else the team shipped. A controlled comparison, where some visitors see the old page and some the new one over the same period, is one way to learn more about a single change.
In the dental software example, the first reading could go either way. If the close rate rises, that fits the team's reading without showing the worry was the cause. If it stays flat or falls, the interpretation may have been wrong, and the buyer conversations are worth revisiting before the next change. The metrics worth tracking and the question of whether a redesign will fix low conversion rates both start from that baseline.