Half halts work well. But you will not get roundness until your horse is working form back to front. Just using your hands is not the correct way to get a round frame. You must use your legs.
I wouldn't even worry about getting your horse round..just get your horse using his haunches, and everything else will fall into place.
To do a proper half halt, get your horse in a nice extended trot, then squeeze with your calf, and close your hands around the reins for three seconds, then release the pressure. When your horse responds praise him..and try again. Then once he gets it in the extended trot..try it in a working trot.
You should start to feel your horse working from the back, and he will become softer in your hands.
Another thing to get your horses hind end stronger is a lot of transitions.
I was agreeing with you until the highlighted section.
This is not a 'proper' half halt, and I also believe your reference to an 'extended trot' is meaning a fast or bigger trot. This is not an extended trot - don't get it confused.
I wasn't going to say more as this topic has been gone over multiple times, but don't want to OP to become confused with aids.
The SEAT is the most valuable aid we have, it forms the largest section of our body - upper leg all the way to your head - therefore have the greatest influence. If a horse can feel a fly on its side, it can certainly feel a change in your seat.
A half halt is primarily a seat aid, and is difficult to perform if you are not entirely sure of how to give one, and the horse is not yet trained to respond immediately.
Before telling a person to use half halts, they must know what the purpose of a half halt it. In dressage, we use a half halt to rebalance the horse, by steadying the movement for a split second, then asking the hind leg to increase its carrying capacity and come further under the horse's centre of gravity. We then ride on the get more power in the pace. A half halt alone will not create a round horse, but is more effectively applied when the horse is already round and travelling.
When training a half halt (yes, you need to train a horse to respond to a half halt, contrary to popular belief that says 'just half halt lots and horse will be round') have the horse must be travelling forward from the leg (absolutely no point in training half halts if responses to basic aids are not sharp enough. Train your horse to move away from a touch of your leg and come back to an engagement of your core muscles first) and willing to come back to an engagement of your core, NOT a pull of the reins.
Starting on a 20m circle, in an active trot, 'stop' your seat from moving with the horse (engage your core) until you feel the horse steady slightly. When you feel the steady, allow your seat to ride on and lightly apply your calf to ask the hind legs to power forward again.
Initially half halts will be performed over a period of time, breaking it into sections to have the horse steady, rebalance, and then go again.
However once trained, the half halt should be performed in just a single stride, and the horse will increase the power and carrying capacity of the hind legs, come over the back and into the bridle, increasing the elevation of the pace in just a single stride.
Back to roundness -
What is roundness? Roundness has extremely little to do with the position of the horse's head. It comes from the engagement and impulsion of the hind legs, swing of the back and willingness to come over the back and into the bridle. Your reins control only the degree of flexion and provide and sensitive, 'live' contact for the horse to work into. Use of the reins for anything else simply blocks the hind legs from coming through - which is probably why you felt that your horse was bracing when you see-sawed OP. A truly round horse has not been forced there. It comes from correct work and a sensitive, feeling rider.
Much like a half halt, the roundness first comes from a horse's reactivity to the aids. Before you even think about where it's head is, you MUST train the horse to be sharp off the aids. A simple touch of the leg to move forward immediately, a hold of your core to come back. Get this sensitivity in all transitions and within paces and you'll be well on your way.
Your position cannot be all over the place when asking for roundness. Slouching, chair seats etc. will not help you at all. You MUST be able to engage your core or you will be ineffective.
At this point, you can begin to introduce the half halt.
From sensitivity to the aids, we then need suppleness and forward thinking. Forward thinking is just an extension of aid sensitivity, if the horse respects your 'go' aids he will be forward.
Suppleness is developed through multiple changes of rein, and the beginnings of lateral work. Leg yielding in walk and trot is a good place to start, as it introduces the motion of moving from the inside leg to the outside rein.
Turn on the forehand is also a very good exercise, teaching both the horse and rider to steady to the seat and outside rein, and step the inside hind across and under the horse's centre of gravity.
If you have achieved all of this and are doing it correctly, 'most' horses will naturally start to work rounder due to the increased use of the hindquarters.
There is more to explain, but very difficult over the internet. You cannot learn to ride from a book or from text on the internet, only get an idea of what to do. Riding experience comes from hours in the saddle. Just like you cannot learn to play gold like Tiger Woods by reading a book. You need to get the 'feel' of the sport.
OP, do you have a trainer? If your trainer is asking you to use the methods you described, to 'round' the horse, please find a new trainer that knows what they are talking about. There are often kids who come onto this forum asking how to 'collect' their horse before their trainer hasn't taught them. Then you find out that said kid is just no where near experienced enough to be worrying about 'rounding' the horse anyway.
Take what has been said here onboard, I hope you have learnt that pulling the horse's sensitive jaw around with your reins to force it into a forced 'head set' is not the way to go, and that it takes a lot more work and skill than that. But please go back to a good trainer and get the practical experience you need to start introducing roundness to your work.