Shanks and Kaido had a skirmish – not to sound condescending, but a skirmish/scuffle is a brief exchange, not a full-blown battle. It doesn’t say Shanks defeated Kaido, just says they had a skirmish. Surely, Shanks is not so weak that he can’t even handle a skirmish against Kaido. The Emperors are individually extremely powerful, a fight between them would likely last hours at least. Could very well be days. They didn’t have an all out battle, but a skirmish.
Then we have to ask what was Kaido’s goal?
Even if we go with Kaido defeats Shanks in a fight, it wouldn’t be a short fight. By the time the battle is over, Kaido would be weakened and the entanglement between Whitebeard and the Marines would have ended. Putting all of these factors together. There was no point for Kaido to fight Shanks for a prolonged period because he’d still not have achieved his goal because Shanks was not going to get out his way unless Kaido wanted a full blown war between his crew and theirs.
Let’s not forget while Kaido is portrayed more as a brute, he’s not stupid. He’s intelligent and that’s reflected in Jack’s character in my opinion. Jack did a great job concealing Raizo and even after dismembering the Minks (which would likely lead a majority into buying their story of his absence), Jack still believed Raizo was at Zou. It shows how intelligent and astute Jack is; he might be juggernaut, but I’m sure the Beast Pirates are not all brute strength, but also are intelligent. Would be nigh impossible to run such a huge organisation without some smarts surely.