I know I’m late to the party here, but wanted to throw in my two cents on the matter.
Knowing that horses don’t lose their baby teeth until halfway between four and five I prefer starting young horses without a bit. The Vaquero tradition used their bosal hackamores and some traditions in Europe would start their horses in a cavesson (much like a lungeing cavesson, but the lateral rings sticking out a little more sideways).
I imagine the H/J crowd would probably give you an odd look showing up in a bosal. I have ridden in them a little, though the one I have had access to has cheek pieces that lay a little close to the eye.
Sidepulls require a responsive horse as there is little reinforcement (and no leverage) with the pressure across the nose (unless you’re using one of the Western halfbreed deals with the rope instead of a flat strap).
I’ve worked with three different variations on the cross-under theme. Dr. Cook’s being the original and the first I played with. My mother in-law has a mare who had lesions in the corners of her mouth that healed crooked so the bit can’t lay straight across her mouth and was the impetus of our bitless adventures (staring with the bosal).
Dr. Cook’s has a release issue, not just from the rein giving back, but a strong lateral aid will cause the headstall to twist on the head and the crownpiece does not right itself.
The Nurtural adds their patentented “circle-X” that prevents both the lateral twisting and increases the speed of the rein release. It also has a “grippy” noseband to prevent it from sliding up the horse’s face.
My issue with both the Nurtural and Dr. Cooks (and their many iterations) is the rein rings used to connect to the cross-under straps. It can be mitigated a little by ensuring that there is minimal strapping past the nose rings, but the sizing doesn’t always allow for that. Not only do you have the weight of the ring, but you have the buckle or snap used to attach your reins to said ring that adds even more weight and, to me, gives a weird, definitely not smooth feel as you pick up and release the reins. Plus on a loose rein you have the rings flopping and potentially tapping the horse in the jaw (had at least one horse not appreciate this).
I do like the concept of the cross-unders and spent a long time trying to find a better solution when I came across Moss Rock Endurance and their “Freedom” and “Evolution” bridles. I tried them out and absolutely love them (the Evolution specifically) and, better yet, my horses go well in them too. I’ve been using them over two years now and have to say my horses are much happier in their work (lesson horses the lot of them except the two year-old of course). I also appreciate their most reasonable pricepoint (reins included even!). They feel much the same as the bitted bridle without the infernal rein rings, they have a functional throatlatch that helps keep the bridle stable, plus they can also be used bitted, as a plain sidepull, and as a halter (which also makes spontaneous lungeing very convenient). After working in these I wonder if part of the “release” issue the Dr. Cook’s has isn’t caused by the weight of the rein rings. Short version: Moss Rock’s cross-under bridle is super versatile and is also very reasonably priced.
I also have a Smart Bridle sidepull from Running Bear that I have used with the Moss Rock reins. It also has the advantage of being a halter, bridle, and can be used with a bit. They are a little more expensive, but they offer more color options and are a bit fancier.
My crew are happy going bitless, especially with my students as they learn proper use of their rein aids. I can swap and ride them bitted as well without issue (my Arab gelding has always been “noisy” with a bit, but he’s a mouthy goober anyway and the only “bit” I’ve had him quiet in was a Stark Naked jaw bit made of biothane strapping).
It all boils down to what your mare likes and goes the best in. If you plan on showing, what is legal in the ring will also come into play. I hope you’ll find something that works well for both of you and you enjoy many years of happy riding with your mare!