It’s even stronger than that. There is good evidence to suggest that, at very large scales, the universe is very nearly isotopic and homogeneous about every point. Most notably, the cosmic microwave background is almost perfectly spherically symmetric, with relative temperature fluctuations on the order of $10^{-5}$.
I think the idea that the universe would be spherically symmetric about a single point comes from a common misconception that the expansion of the universe (the “Big Bang”) started at a single point like an explosion. However, this isn’t how our current understanding of the universe works at all. Instead, cosmological models suggest that the Big Bang happened everywhere, all at once. Here is a very nice MinutePhysics video about the topic.