Most ponies, especially older ponies, are smarter than you would think. They learn quickly, have experienced more, so they react quicker than most horses in good or bad ways.
As the mom, you sound fearful, but working to overcome it. Willing to TRY doing things to help reduce your worries. It may be real hard, but do not project your fears so the child picks up on them. Fearful children have hard lives, worrying about everything. Even things that PROBABLY will never happen.
Being cavalier about injuries after asking or checking for severity, is the best way to handle things with kids. As you say, parents always seem more worried than the kid who is not coddled. Without lots of blood, limbs bent at wrong angles, there usually is not much to get excited about. Husband sees lots of injuries in his work, both human and equine. His line is "Wound is a long way from the heart!!" No much sypathy from him because he sees REALLY bad injuries regularly. Not even all horse related!!
With equines large and small, you will pick up some bumps and bruises along the way. Part of being with herd animals who use their body to help with "status", leadership in a group. Group could be you and horse, someone HAS to be in charge. Horse will be the leader, if YOU don't take command!
I would second you getting some riding lessons, along with horse handling lessons. Knowledge is power! The more you know, the better you can control conditions around you and any horse you handle. Your daughter should be getting this safety and handling information in her riding lessons. She is helping prepare the ponies for riding lesson, grooming and unsaddling, after lesson care isn't she? If not, I would ask that she did get it added to her lessons.
Those marvelous horses who have "been there, done ALL that" are a gift for those who get to use or own them. Often passed on, not sold, from family to family. I have heard of some folks finding them with advertising, but not common. They teach the whole family how GOOD horses should be, set expectations very high for horse number two! May not be pretty, but worth the money if you ever come across them while looking for a horse.
I put up with some not-nice animals when I couldn't afford better. They just had petty reactions to anything you wanted of them. No more, I won't put up with that now. We may start with young horses, but they all are kindly, nice minded and enjoy being with you, whatever you want to do with them. Their first reaction is NEVER to kick, even if surprised. Anyone here who kicks, goes on the retrain a kicker program. If we can't fix it, they LEAVE. This would be kicking in the herd, kicking in the barn, as their first startle reaction when something happens.
As mentioned, you can visualize all the BAD stuff happening. All it does is give you a headache or an ulcer. Life is for living, not worrying about possibles! You take reasonable care in your activities, try to pay attention to your setting. Not much more that you can do to prevent things happening. I also look at "accidents" and setting or what was going on when it happened. Do I need to change how I do things? Maybe the setup is poor, causing "almost accidents"? Those near miss things are your warning to CHANGE something!
If you want to worry about safety, helmet is a BIG DEAL. You set the good example for child, by wearing a helmet EVERY RIDE. Helmet needs to fit you well, be comfortable. I would say to get your own and one for the kid. Should be an expected part of yours and her riding clothes, NO EXCEPTIONS. You NEVER know when you or she might fall off, so you and she ALWAYS wears the helmet to ride.
When buying check inside the helmet for the manufacture date and ASTM or SEI Approval for horse activities. This date is when helmet was made, should be less than 5 years from that date. Lots of helmets on sale that are OVER the date, should be discarded, but they can't lose the money invested. So I would not purchase a brand new, in-the-box helmet that was over 5yrs from the maker date. Helmets with no date, no Approval sticker, over the date, are merely HATS. Hats are not protective, just items of apparel that cover your head. Not a Safety item at all.
You should purchase a NEWER helmet when the date inside expires at 5 years or helmet has gotten hit hard when someone fell off. Stored in hot cars, trailers, also is not a good idea. This is manufacturer suggestion, plastics deteriorate and impact material inside develop tiny cracks or impact material has been "used up" with a crash. I would not be buying a used helmet, don't know how they kept it or if THEY crashed wearing it.
The saying "A helmet for every ride" is a good one to keep in mind. Falling off can happen when you get on horse for ANY reason, so helmet has to be on rider's head to make a difference. Doesn't matter your skill level or experience, horse can put you off or fall down, you will hit your head.