If you are not looking to make it to the tip top of competition, it doesn't matter what type of horse you have. With enough hard work, perseverance and the right training, nearly any horse can perform passably well in nearly any discipline. As has been said before, most western horses, particularly those that work with cattle, or in speed events, tend to be smaller, and more compact. They have to be built like little sports cars to maneuver like that so easily. A draft or a draft cross is not incapable of participating in the same events but they are unlikely to win much. Drafts and draft crosses have a disadvantage in the pleasure arena as well. Judging in these disciplines is more subjective and judges are likely to mark down horses that do not fit an ideal type, even if they are performing well. What matters most is that you find a horse that you love, that has a temperament and level of training that suits you, and that you have fun with whatever discipline you choose to pursue. It's not about the ribbons you win. It's about the good times you have with your horse!
I have a Shire/Thoroughbred cross that looks like a Shire. The advantage to her mixed pedigree is that she has the the good bone, and striking looks of her Shire parent with the athleticism of her thoroughbred parent. She doesn't move quite like a draft. She's much lighter on her feet, and moves more long and low than her purebred draft counterparts. She can be a bit lazy at times but she's not pushy or rude at all. She's very light on a lead and sensitive to her rider's aids.
There are plenty of drafts out there that are not so light and easy to work with, or that are built like train wrecks, but there are drafts and draft crosses that are well-built, and sensitive. You just have to find the right one! Don't let the nay-sayers discourage you from finding your dream horse, but remember to take off the rose coloured glasses and look past those pretty feathers and big feet when you go to look at horses. And bring an experienced friend or instructor with you. You want to find a horse that is perfect for you in every way, not just in size and appearance.
Here's my draft cross. I have never had her in western tack, but my dad runs poles and barrels with his appaloosa mare and I ride with him. Ursula won't beat his horse in timed competition but we are still able to have fun with it!
The one usually western event that she is getting pretty good at is obstacle trail. She can maneuver obstacle trail courses like a horse half her size!