Yes, **depending on how & why it's done**. I don't *generally* like the MR type style of round penning & think it's based on mistaken ideas...
Firstly lets look at the principles behind it. It is supposed to replicate a lead mare's behaviour towards another errant member of the herd - usually a testy youngster who hasn't 'learned his manners'. Horse bands, especially in the wild, get their security from being part of a close knit herd. When a subordinate horse 'misbehaves', the lead mare will often drive the horse off and keep him out of the herd, until he shows his 'sorry face' and is welcomed back in. The errant horse desperately wants to be among the herd, because not only are his mates & family there, others he has a close bond with, but being on the outside is dangerous when there may be predators about.
Now lets look at the behaviour of a hunted horse & it's hunters. Depending on the predator's style, the horse may be pursued relentlessly until it gives up in exhaustion. Or the opportunistic hunter may close in when it is trapped in a blind canyon or such. In either situation, to begin with, escape is obviously in the forefront of the horse's mind. But when it begins to become exhausted, or is otherwise trapped without hope, it will eventually 'submit', 'shut down', 'give up', whatever you want to call it. It will slow down, lower it's head, lowering it's adrenaline, & often even turn towards the pursuers(very similar behaviour & 'signals' to the 'sorry face' of first scenario). It doesn't tend to do this unless completely spent, physically & emotionally. The 'shut down' phase is suggested to be to basically allow the predators to get on with their job without causing the prey(meal) any further undue stress & pain. Also it saves what faint reserves or hope that may be left for any last ditch surprise escape attempts.
So, IMO, after reading the opinions of ethologists & behaviourists who have also studied & written about it, it seems to me that many styles & principles of round penning are a lot more in line with the second scenario than the first - the horse is placed in an enclosed environment it can't escape from, with a human that it may be terrified of, who is focussing on & 'chasing' it. Even if the horse doesn't perceive you as a predator, he likely doesn't see you as part of the herd, let alone his respected leader to begin with, likely doesn't naturally feel the need to be in close with you & the danger of being on the outside - especially as his true herd is probably out there somewhere - he may or may not be bonded to the chaser already and he may or may not have committed some 'misdemeanor'(or if he has, associated it with the 'punishment').
Now, so saying, I do think 'join up', 'round penning', 'the catching game', whatever you want to call it is a handy tool in the right situation, and there are also many ways of doing it, in a much less confrontational, threatening way. Behaviourally, it obviously certainly can *help* teach the horse to come when asked, follow you and potentially *reinforce* the bond & trust between you. I don't believe it of itself actually creates an emotional bond or trust. It can desensitise the horse to being 'chased', so the horse *responds* with understanding to the pressure, rather than *reacts* in fear. Done badly tho, it can cause the horse to fear the chaser & his tools all the more & become more reactive, less responsive.
With a horse who is already reasonably comfortable with me, I might use a similar technique to MR style to teach him to come when called and to follow me. I also use 'driving' from behind to reinforce my position as leader - you move when I tell you to - and 'lunging' to teach/reinforce responding to implied pressure/cues at a distance. I tend to only use a roundpen or that small of an area for lunging, and sometimes for initially teaching a sceptical but unscared(ie 'dominant' or used & abused type maybe) horse to 'be caught' - ie come when cued.
I wouldn't do it with a scared horse in a tiny area such as a round pen, or so directly (aggressively?). With a wild/scared horse I do it in a small paddock or very large yard - say the size of an arena. & I only do it in that small an area to save time/energy on my part
I also only get after the horse if he turned his back or walked away from me. I 'get after' him only calmly at a walk. Basically, I follow at a distance as he moves away from me - keeping light(but assertive, consistent) pressure on him - but the instant he slows, turns an ear, whatever the *smallest* signal that he's thinking of hesitating, or even just changing tactics, I will stop, relax & walk away a bit(remove pressure to negatively reinforce his behaviour). I will repeat this until he understands that to get me to quit hassling just requires him looking at me.
Depending on the horse & his experiences, I may then start the same game but keep the pressure on until he actually turns towards me & then until he starts to move towards me, but generally with a scared/wild horse I won't try to go that far yet, or even approach him necessarily. I will instead go sit down & focus on something else - say reading a book, plaiting some leather, whatever. I will wait for the horse to approach me & ensure I'm not the least threatening - focussed on catching him, whatever - when he gets there. I also like to have a Good Thing for him to find when he gets there - pellets, a carrot, vine leaves, rosehips, whatever is a treat for that horse. I often won't approach him or push him to approach me until he's already learned to be comfortable in my presence.
Anyway, that's my take on the subject. Interested to hear all the other opinions!