Riding a hot horse

I have a mare that's very 'hot' and excitable under saddle but she isn't spooky and would never dream of trying to run back to the barn etc
She is perfect where groundwork is concerned so I see that as a waste of time for you with this horse.
You have to learn to be a very relaxed rider - I had a lifetime of fizzy gymkhana ponies so to me she was no problem, I love her the way she is, you either like this sort or they terrify you and if they terrify you you will send them into a panic as they need cool and calm
I have long perfected the 'sack of spuds (Potatos) tied up horrible' style just for these sort of horses.
If you use more than a hint of leg it will be off but you need to keep them into the bridle on a light hand - any hint of a hard pull and that forward motion has a nasty habit of turning into an upwards one.
One point - is this horse 'hot' or is it fractious?
You might want the owner to look at possible pain issues - like in her back or maybe even ulcers.

In response to te ground work bit:
I ask because the owner stated she's not done any ground work with her horses since they were broke to ride. In my training as a rider, my instructor would have us do 5-10 minutes of ground work during our practices before a ride as part of our training as horse people. A previous instructor also advised that I do a quick lunge before riding any horse I am not familiar with. Once I even got yelled at for not doing it when she had sold my lesson horse at the time.... I know for a bit that I will do ground work with both horses [about 5-10 minutes before each ride] until I feel confident they respect me. Or does this seem redundant?

I don't know about the gelding, but the mare the owner says does get lame every now and then from being excited. However, the hot gelding is on the bottom of the totem pole so I'll ask and see if there's possible pain injury.

I don't ever think ground work is a bad thing..I thought you were meaning to try to fix the "hot" behavior from the ground and that's just not going to work..Ground work can be beneficial for any horse, as long as you keep it different and not so routine. They get a lot out of it, respect being one of those things.

Ground work is a good way to get to know a horse. 5-10 minutes is not going to hurt anything, it's doing the same ground work, for 30+ minutes a session, several times I week that causes a "sour" horse.

Never heard of a horse that was lame from being excited??
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No, no, I wouldn't have thought ground work would'a done a bit of good for a hot horse as it's mostly just learning to work with them in a saddle.

As for the gettign excited and becoming lame afterward, I may have stated that wrong. The mare [the only mare] is an ex race horse and often gets lame. I think the owner said because she sometimes reverts back to being a race horse, but I am not entirely sure. Her hooves do need work though, and I do know the farrier is going to come soon.