I agree w/ several of the posters here - both for and the one against (Holly).
The mini halters meant for smaller minis (commercially made and purchased) haven't fit any of the horses I've tried using them on (smaller minis under 34" in size). They are made from larger rope and I've found them cumbersome and that they don't work effectively. Better there with a nylon or leather halter - and these days you have soooo many choices in the mini sizes/colors that that's the most fun, cost effective way to go. I really don't have smaller minis (my foals don't stay small that long!), so can't really address the being taller/halter shifting or not working right... I can see that happening though, but I've had it happen with nylon and leather ones as well as with headstalls/bridles.
BUT on larger minis - I've purchased name brand Diamond and Clinton Anderson one's that fit the larger 36" to 48" minis, shetlands, Shetland Xs and Arabs and LOVE them. The name brand ones DO wear out - so don't let the hype about them being "longer lasting" get you... Not true. Same as any other leather or nylon halter - depending on quality of rope and how it is used/taken care of. The weanling size fits my larger ponies, the foal size fits the smaller ones (not any of the foals though), the yearling or Arab size fits my horses. The smaller the horses' head - the bulkier the halter as the diameter of the rope doesn't change. Other name brand halters that I've tried aren't small enough.
Both myself and my farrier have made several for me that I use. I've braided them out of haystring (I REALLY "likey", but whew - time consuming to make), made them out of straight paracord (not braided - too thick that way), and out of a larger diameter, soft rope. I don't particularly care for the real soft ones - for me, the whole point of the halter is instant release of pressure when a response is achieved and that doesn't happen with the "soft" ones. But the soft ones do work for just holding them, IF the horse is well mannered. IF a pony or mini pull back in the soft ones the tie knot over the crown pulls so tight they can't be undone - I've had to cut them off. Haven't had that issue with the stiff ones - not even on a 1,200 # horse that sat back so hard she lifted her front end completely off the ground (she was a regular "halter puller" and broke many regular halters, lead ropes, posts and even flipped a 2 horse trailer over, before she came to me)....
Horse dot com used to carry a yearling size that fit my larger ponies and Arabs for only $7 - that was GREAT. They don't carry it any more (only average size - which is WAY too large for any of mine)..
I like tie halters simply because I LOVE equipment that is all one piece w/ little to no hardware to break. I love adjustable sizes. I've turned my belts into emergency halters when needed (and hate it when I see a loose horse, not always ours, and no belt). The rope halters "mash down" to fit in a pocket of a coat, or in a small car (always have some in all of our vehicles now - even hubbies!!).
In the end - all halters are a personal choice. Any halter of good quality material & good fit, is "pricey". All halters can be made of substandard materials - though they sure do look great in catalog/online pics and even when you first get them... Rope halters - the smaller the diameter and the stiffer the rope, the "harsher" it can be - but for training that's exactly what I want, to achieve an instant release. Even poor quality rope halters won't break if a horse gets hung up in them - hence why they should never be left on in a loose situation (but then no halter should be, and I have been guilty of that w/ a few hard to catch ones)...