Why should LoL adopt this system? Because too many people are locked into this "lane" ideology, that the way to win the game is to win your lane (win lane, lose game syndrome). Lets take soraka as an example. If someone were to pick soraka and play her in the mid lane (position 2), she is getting a solo lane's worth of farm and exp. but what is her role in a teamfight? unless you have 3k hp with massive armor/mr, you will not have the survivability to stand in the middle of the enemy team spamming ur q to maximize your raw damage. in most situations, even if you are playing soraka mid, you will be most likely staying behind ur "tanks", and ur main task in teamfights is to use ur heals to allies that need them as well as silencing high priority targets (ie cassiopeia/ryze). now lets say the support is a malphite. the malphite will mostly be used for his ult engage. however, there is very little damage associated with the engage, so malphite would have to rely on his teammate (namely soraka in this example) to actually do the damage necessary. Lets review this hypothetical situation. Soraka still needs to be very close to the enemy to do any real damage, and this damage takes a while to get there. malphite provided a great initiation initiation that cannot be easily abused by the mid soraka. in most situations, the gold/exp from the mid lane is better used on malphite instead of soraka since his raw damage from building ap is much higher than soraka's. You could have very well dominated the mid lane as soraka, but you would still lose the teamfight if you cannot get close enough to spam your q at the enemy. Of course, you can argue that the heals make up for the lack of damage. However, you cannot guarantee that the damage will come from your team (missing the adc, team gets caught, team comprises of low damage, support oriented champs). Thus, in this situation, the 2nd position is more or less wasted on a champion who may not do the necessary damage needed in a teamfight.
The example I gave is very extreme. Most of the shuffling in position is between position 2 and 3 (typically the mid and top laners). Lets say you have a gragas mid and jax top. who should be prioritized for gold/exp. purely from the "laning" concept, the mid is the "carry" and the top is the "bruiser". This is implied that the mid champion will do more damage for the team than the top can given the items/levels. Most people would agree that jax needs a lot of end-game items to essentially 1v5 the enemy team. Gragas, however, only really requires a deathcap/void staff/sorc boots to be happy when tossing barrels/ults. The jax would take position 2 in this setup while gragas takes position 3. An example of where position 2 would be given to the mid is when there is a champion like karthus, since there is more guaranteed damage coming out of a farmed karthus.
What about the jungler? Well, it depends on what jungler you are playing. If you are amumu, the only thing ur team needs of you is to bandage toss ur way into the enemy team and lock down everyone with ur ult. you can do this with purely levels were it not for ur fragility without hp/mr/armor. Thus, it really does not matter how badly you got shutdown by invades as long as you can make clutch ults happen for your team. Sometimes, you see junglers like master yi (famously atlanta). Team comps that have these kind of junglers require the jungler to take a higher position on the team.
(montecristo analyzing atlanta's jungle yi)
If you look at the team comp, we have kennen top and maokai mid. Kennen went the standard WotA while maokai went straight tank. But of course, someone needs to tank right? If that were the case, why isnt maokai in the jungle and let master yi take a solo lane. after all, it is more gold for yi. This is a prime situation for the 1-5 system. Maokai (position 3), with all the ap he can get, will not outdamage master yi (position 4) when given the same gold. Absolutely cannot. Maokai needs the initial gold to get to a tanky position for himself, after which any more gold invested in tanky items is the cherry on top. This is where the gold focus is given on master yi. In the midgame, maokai essentially goes around with the team trying to gank the enemy. He is now tanky enough to dive into the team, causing havoc, and coming out alive even without his ult. In the meantime, master yi comes in and cleans up teamfights. The gold priority has shifted from maokai to master yi. In other words, anything that master yi could do to get more gold will be more impactful than anything maokai could do to get more gold. In this case, the way to get master yi more gold is to take the global objectives as well as killing enemy champions. This is where master yi takes the 2nd position over kennen and maokai. The more gold master yi gets after the laning phase, the more impact he will have on the teamfights.
But this is all pretty obvious, isn't it? And yet, people do not realize the true impact of this system. Lets draw up a hypothetical situation. Say the team comp in question comprises of an irelia top and a karthus mid. Irelia managed to get a double kill on top early due to a failed gank attempt. She now has a tremendous advantage over top. Karthus, however, has had mediocre farm and is getting pushed around by an aggressive mid. By 15 min, irelia has a full triforce along with a t1 boots. karthus only has a catalyst/sorc boots and is saving up for the blasting wand. What should the team do to solidify this advantage for irelia? Does she stay in lane and continue to bully the underleveled enemy top lane or sacrifice her farm/presence at top lane to help karthus? In most situations, she should rotate mid so that karthus could have an easier time getting the farm he needs (keep in mind that by rotating out of your lane, you are giving space to the enemy top lane to farm up and get levels). This could come from kills/tower pressure/contesting enemy blue/dragon. Because irelia will not get substantially stronger after the triforce, her priority should be helping karthus get farmed up. Even though it seems irelia's impact on teamfights is much greater than karthus, karthus still benefits more than irelia given the same gold. This is because irelia experiences an upward-sloping plateau in terms of teamfight usefulness.
How will the dota 1-5 system help lol players? By reminding them of who the carries are and how to get those people more ahead. A darius should be happy to forfeit some farm or even lane if it means he can help snowball a hypercarry like kogmaw. A katarina should be basically camping for jax to hasten jax's lategame potential. The 1-5 system should help some players elevate their play by focusing on what is important for the team composition and how to best play their role to help the team win.