The gameplay doesn't really change over time. Many of the upgrades are just economy upgrades that change nothing about how you play, and any exceptions have really small effects. Even the dogs don't really change anything since you can't trust them to go for a given sheep. At best they reduce how many you miss.
In fact the dogs might be a little bugged with how many sheep they'll ignore. I don't know if there's a deliberate cooldown or what.