I agree with the other post-ers-- it will depend on the stallion and the handling, training it has had, as well as its natural inclinations. (Some stallions, even with the most experienced training and handling, are tough to be around.)
I have owned several stallions including an excellent Arabian stallion, and some excellent Appaloosa stallions. I would not buy or keep a stallion around that was not gentlemanly (having a basic kind willing nature before training, and being mannerly after sufficient training-- I have experience handling stallions for both daily maintenence as well as breeding and some behaviors can be taught, refined, controlled, or if taught wrong, some things can be "undone"-- other things are part of that stallion's nature or have become ingrained habits and usually "go with him" no matter what training he is given) and my Appaloosas were every bit as nice to handle as the Arabian-- in some ways quieter.
If you are not going to have your own mares to breed nor stand a stallion to the public, I would not keep an intact stallion around just to breed a mare or two for a previous owner. Even the nicest stallion has specific housing and handling needs, and even the mildest kindest stallion still has a strong built-in biological drive to seek out and breed mares..... IMO it would be more risk and trouble than it was worth to have a stallion if your main purpose was for a riding horse and trail partner. Plus, as others have mentioned, some events and trail rides do not allow ANY stallions.
As far as Arabian stallions being shown by youth-- meh, no biggie. Appaloosa stallions are allowed to be shown by youth in Open classes at Appaloosa shows as well. They used to be allowed in youth classes, but that was changed long ago mostly because there are a variety of ages and experience levels in a youth class, and not every youth should ride even a great stallion-- also other youth might not know what to do if their mare reacted to a stallion-- even if the stallion is totally mannerly, some mares KNOW a stallion is present, and the mare can be the troublemaker when she thinks a stallion is "available".
There are other breeds where stallions can be shown by all ages as well-- the Arabian horse assoc. Uses USEF rules to govern their shows, but not all other breeds use USEF, so saying Arabian is one of the only breeds that USEF allows youth to show stallions is misleading, since USEF doesn't make show rules for the majority of breeds anymore.