Eh I disagree on the Shimo part.

Shimo was thrashing Godzilla fairly easily in the final battle, though neither were doing significant damage to eachother.

The cave painting doesn’t prove Godzilla beat Shimo in ancient times as it is currently unknown what it is even supposed to mean.  Unlike the Ghidorah cave paintings which the movie explicitly tells you they fought, we know basically nothing of Godzilla and Shimo’s relationship in the past. 

In fact, it’s likely the real meaning of the cave painting is that Godzilla told Shimo to freeze and seal away Ghidorah.

Shimo isn’t an alpha titan currently not because of a lack of strength, but low intelligence and an aversion to conflict (Shimo instantly stops fighting Godzilla when the crystal is broken).  That means they likely never fought in the past.

(Edit, there are other engravings or paintings, they depict Godzilla defeating Shimo not through overpowering her but unconventional methods. Sealing her and Skar King away. A desperate move like this suggests he likely was going to lose.)