The difference between the chess and go skill patterns is because chess and go have vastly different algorithms.
The chess skill changed linearly because the algorithm is easy to compute by point values (finding the paths with the most pieces compared to the opponent, or positions leading to this), and modern algorithms aren’t much different from early ones. In other words, taking the enemy’s queen without similar cost is always extremely good (if the computer can look far enough ahead to check for traps), and computers are mainly limited by how many turns ahead their processor can look.
Go, however, is far more subtle, with pieces being drastically different in value based on what occurs without them, and a few pieces in the wrong spot can lead to the loss of a quarter of the board 40 turns later in a subtle way, such as providing a ko threat or dead shape. Counting the territory in 5 turns is near-useless without considering how each piece interacts with all others. In this case, the limiting factor is not processing power but algorithm design, and the rapid gain happened because of insights in algorithms.
What determines whether AI development will be sudden or gradual will be which type of limiting factor it has. Self-driving cars had a big jump then a stall because it is an algorithm difficulty. Computer graphics improved gradually because it was a processing power difficulty. Sentient AI could be like one of these, or have a different limiter I haven’t thought of, but whatever the limiting factor is would determine the rate for each thing.
I’m not an expert in AI, but am very good at chess and go.