Is the age of the Universe really 13.8 billion years?
The age of the universe is not calculated based on the size of the visible universe. The age of the universe is being calculated based on the fact that the laws of nature have no direction. This means that you can use the laws of nature to predict future behavior, but also assume previous behavior. Based on calculating backwards with the laws of nature, for instance, general relativity, physicists can to a pretty certain degree calculate backwards how the universe started.
The fact that the universe is expanding and the size of the universe is what it is, has no relation to the age of the universe. Please, look at another video from Don Lincoln about the size of the universe and its age. Here it is explained that the universe already had an extra large expansion making it bigger than the age.
What will happen, though, is that the visible universe, which is smaller as the actual size of the universe, will be more empty over the course of time. In the 5 billion years you mentioned, there will be more galaxies outside the visible boundaries of the universe, because they are moving away from us faster than the speed of light. The expansion of the universe, hypothesized due to dark energy, makes the universe bigger, not older.
If the laws of nature don't drastically change, and we use these laws in 5 billion years to calculate the age of the universe, we should end up with the right number, 14 billion plus 5 billion years.