How did early modern palaces etc keep lovely lawns fit for a king? Scything seems too rough. And grazing sheep or goats seem like they would bring too many problems. On my hands and knees with a sickle or shears? It all seems like a pain in the arse. This is to settle a pub argument. So it is highly important.
Ah, tennis, the sport of kings! A question I am somewhat qualified to answer. My citation for most of this is from The History of Tennis: An Authoritative Encyclopedia and Record Book by Bud Collins, Third Edition, 2015. Collins was the single most knowledgeable source about the sport in the world, and sadly passed away in 2016 RIP.
To understand lawn tennis, you have to understand the history of tennis, and how it began. Well, in the words of Reb Tevya, I'll tell you...I don't know! The truth is that nobody really knows the truth about the history of tennis, and it's probably a lot like pizza: everyone claims to have invented it, everyone has stolen each other's recipes, and everyone claims that their version is the best.
To jump forward a bit though: the gardener had a really easy time maintaining Henry VIII's tennis court, because they were indoor courts for what is known as Real Tennis, and outdoor Lawn Tennis wasn't really a thing yet. Hampton Court and Henry VIII's palace have some of the oldest Real Tennis courts, dating back to the 1500s, but more resembled squash or racketball than what you and I would consider to be modern tennis. Our knowledge is fuzzy, and the history of it is unclear. We think that back in the early days of the sport in the 1100s-1300s it may have started off with monks essentially playing handball in northern France. It would have been played with their hands, or gloves, with a deerskin ball, indoors. The sport rapidly migrated to England and Henry VIII built courts in his palace, and Cardinal Thomas Wolsey, Archbishop of York, Lord Chancellor of England, was a huge fan of the sport as well, and contemporary of Henry VIII, so he ordered the construction of courts at Hampton palace. But again these were Real Tennis courts, indoor, with no lawn to be seen.
It is thought that the sport migrated outdoors in the late 1800s. The All England Lawn Tennis and Croquet Club Association Championships at Wimbledon, for example, have been held since 1877, and it is the oldest tennis tournament in the world. The Club had originally been founded as a Croquet club, because the lawns only needed to be flat and relatively free of obstacles, not immaculate for a true bounce required in tennis. According to Wikipedia, the migration from Real Tennis to Lawn Tennis in the 1800s is almost definitely because of the advent and patenting of the first lawn mower in 1830, which allowed for much more precise control of the grass and grounds, and much easier maintenance of course. Wiki is of course not a primary source, and I couldn't corroborate it anywhere...however it does seem like a very reasonable assertion.
So to answer what I think is your main question, about how lawn maintenance in general worked, it would have been an enormous pain before the advent of a lawn mower in the 1800s. I can't speculate on how they did it, but hopefully you know a bit more about the history of tennis now. Enjoy the pub!