@CrealCritter , I will have to TOTALLY and EMPHATICALLY VETO any attempt to milk from the other side of the gate... mostly for the reasons stated with the possibility of getting an arm hurt or broken, or a cow kicking the gate and injuring their leg or the person behind it., It is not safe to do it that way... Sorry to throw a bucket of cold water on the idea.
You need to get them to come into a head catch type chute... width depends on the animal and I will make it a point to measure the width of our chute at the barn tomorrow. Then you close the head catch on their neck while they are eating some grain... and you start with touching, rubbing, brushing them. Then move to the udder area and rub and gently touch the udder... They have got to trust you to allow for milk let down. This needs to be done in advance of them calving and all so that once they get the idea that there is feed in there... that is what it is for... and milking will become a part of it.
Since they are feeding the calves this is something you can get them to learn to do now... to be fully comfortable with them eating in the bunk, and you walking around them. Then when you wean off the calves and dry them up, you can pick the one that is most comfortable with the human presence, and get them comfortable with the closing/locking of the head catch... and then totally used to someone being around them while they are eating.
I am thinking that you are NOT going to try this, this lactation with them? The calves will stay with them until ready to wean and they should be dropping way off in production and be ready to be dried off.
Tying them with a halter is not a good idea unless they are used to being led with a halter from a young age. I use a neck chain with my one holstein/jersey cross... but she had been milked on a dairy after her first calf, so the next lactation was a no brainer for her... she does not like the chain I hook her to in the barn but once she knows she can't back out or anything, she accepts it and stays pretty still... I do not use a stall for her because I am usually grafting another calf or 2 one her so need access on both sides for the calves to go on her...
Calm cattle in the field is one thing,.... being in a barn, in a head catch, and being fooled around with when they are fresh is quite another... I honestly would not be doing more than totally taming one down now to accept being restrained in a head catch because they are only going to want to let milk down for the calf... and it could be a scenario of someone getting hurt...
You NEVER let them out of a head catch if they are fighting it... and never let them get loose from it... or they will try it every time after that... they have to learn that fighting it will not accomplish anything... and that after they eat and all, you will turn them loose when you are ready. Hence some of the TRUST..
I appreciate your wanting to milk and all... a milking machine and all the cleaning will take more time to assemble and disassemble and clean than the time to milk... and I am not saying they are bad... but it is a big investment for no more milk than you will get off a beef cow. and once you start, it is tricky to share with the calf... and a beef cow will not make that much extra so the calf will get somewhat shortchanged for milk unless you wait for it to be older and do a 12 hours on/ 12 hours off with the calf and then she may not let her milk down for you. Beef cattle tend to be a little more "notional" about milk sharing.....have had several 1/2 beef 1/2 dairy animals over the years... and only some of them take to it.
You cannot properly clean the udder, attach the teat inflations and monitor a milk machine from outside the fence or gate. NOT possible... sorry to burst the bubble... but it is an up close and personal operation...and once you start the milking, you have got to stick with it or the cow will either not let her milk down for you, or she might hold it some and could get mastitis from the irregular schedule/times of being milked.
I am not trying to be critical or discouraging... but milking is something that you can only do so much to be careful and safe and you cannot do that behind a gate or anything that could hinder moving away quickly, or that an arm or hand or leg could get slammed against or caught between or kicked into the gate if the cow takes a notion.
There are chutes that can be totally opened from the height of the cow's belly to the floor for the express purpose of allowing a farmer to get an animal in a chute and then work on getting a calf guided to a teat... no metal or hard or solid objects to get a calf's body caught between a kick and the framework..... so they can't get caught between or against a gate or something that won't "give".
If a cow gets her leg through the bars of a gate they can break it sooo easily... and it does not have to be a big opening between them... can't do that with a solid gate... can't reach through for the udder with a solid gate either...
I will measure the width of the chute for your general knowledge... and in case you need to create a chute to restrain an animal that needs the vet to work on it for any reason...and you should have a regular chute and a head catch for any emergency vet work that you might have to do... but it is not for milking purposes... It is also easier to prevent a cow from kicking if you are close with your hands/arm underneath her on her udder while hand milking...
Some chutes have 1 bar that goes from front (head catch area) straight back to the back of the chute, horizontally, at approx mid belly height, to keep the animal in the chute but allow full access to it's side for any work to be done while not allowing the animal to swing it's rear out and around... but having control of the side to side movement.