To be fair, I think most people do not dodge to punish you if you do not trade. They rather have a champion they don't want in the first place and no rerolls. If everyone with a champion they want declines the trade, they dodge, but the incentive is still "I don't want to play <boring>" rather than "if I don't get to play <exciting>, then at least I'll make sure you won't either". The latter "vengeance dodge" is far less common than the former "aversion dodge", I'd say.
Dodging in ARAM in general is a mentality problem. As the name suggests, ARAM is about playing random champions. But those people don't want a random champion, they desire a limited pool of champions a priori and only then want random choices within that pool.
Because of that faulty attitude towards the game mode, I go as far as to impute that they don't actually want to play this game mode in the first place. And therefore, in my opinion dodgers should be blocked from the ARAM queue for a much longer period of time. Even very extreme penalties like 24 hours seem justificable, because since according to my reasoning, they don't want to play the game mode anyway, so we might as well prevent them from interfering with those who want to embrace the game mode and its implications.
It's like going to the learner driver training area and commit reckless driving. Apparently you're not interested in the services this facility is meant to provide, and even worse, your behavior is interfering with those who do want to use it in orderly fashion. As a consequence, you'll most likely be expelled from the area and won't be allowed to enter it again in the future.
The problem of dodges in ARAM can be approached from different sides, each tackling different aspects of the problem.
- My method tries to penalize dodging more severely to reduce its appeal. Especially the first dodge is a positive trade (like in every queue): "get penalized for 15 minutes to prevent 25 minutes of unpleasant game play". It also passively decreases the number of dodges by locking out those who still dodge for longer periods of time, reducing the number of times they can exert this behavior in a given amount of time.
The problem with that approach is that false positives, i. e. those who actually lost connection or had to quit because of a technical problem will receive a very harsh penalty for doing something that the system actually would ideally not like to penalize at all. (Note: the naive suggestion to introduce free dodges per time is a bad idea, because it suggests not that "dodging is sometimes okay", but rather that "dodging sometimes is okay".) Secondly, if we assume people who dodge in ARAM don't care about that game mode, going down that road means we'd also have to assume they don't want to play it as often, hence a longer block from that queue doesn't hurt them as much as we'd want it to.
- Your method tries to mitigate the harm dodging imposes on the other players, hence making dodging a less painful problem for those who don't do it.
The problem with this approach is that it requires not only a different champion select, but also a whole new queue mechanism, one where the information which champion you have from the previous champion select has to be taken into acccount. Right now, the queue can simply put 10 people together. Since everyone is guaranteed to have a sufficiently large champion pool, there's no possible way to combine players in a way that they cannot form a champion combination without duplicates. But now the queue must neither pair nor match two people with the identical "locked" champions, hence the algorithm gets more complex and queue times can increase. Furthermore, if dodging is less painful for everybody else, in general it becomes more accepted ("if you want to do it, go ahead, it doesn't bother me"). It is questionable if that is the message you want to get across.
A slight variation to your proposal is to store the champion for everybody including the dodger, so that dodging in general doesn't have any benefits over playing it. This has the same problem with queue complexity, but might be the most effective way to abandon the biggest incentive of dodging.
- And finally there's other methods that try to prevent dodging from happening in the first place by alleviating the urge to do it:
The first proposal is to allow for more rerolls, for example by allowing to stack way more than two. In theory, this can allow people who are "usually okay" with the random champion to get around dodging in the case they get 3 champions they don't want in row. The problem with that approach is that it only delays the problem for many cases. The pickier a player is, the less likely a given amount of rerolls is sufficient to give him a comfortable pick. And at some point, if the number of rerolls gets too high, the aspect of randomness begins to suffer.
The second proposal is to give people a random choice of multiple champions, and allow them to pick one of that, reducing the chance of getting stuck with an unfavorable pick you need to reroll. If you get provided with 3 champions and then can reroll them individually before chosing, you can try to reroll the least favourite and if that gamble didn't help still go with an "okay" pick. The problem with that approach is that it obviously raises the minimum champion pool requirements by the factor of the number of choices, because the individual choices must be unique for everyone.
The third proposal is to allow people to manage a list of champions they do not want to play in ARAM. The list could have different restrictions, obvious ones such as the minimum number of champions that have to remain open, but also a possible maximum length to restrict people to disable only those they really really do not want instead of allowing them to narrow down the list to the ten cherries in their roster. The problem is that this can limit the diversity in ARAM, and it takes away the aspect of "discovering a champion you're not that comfortable with". But on the other hand, many argue that often the same result will be obtained because people are dodging, just with more pain for everybody else.
And finally there are many more proposal, most of which revolve around how people pick their champions to reduce the likeliness of getting stuck with a champion you don't want. Those do of course have their own downsides.
As you see, there's a lot of different thoughts on that problem, a lot of proposed solutions, each with different problems and considerations you have to make. I agree this problem is something that should be looked upon. Alas, I afrain this isn't something that will be looked upon, as with all the problems in ARAM, since it is ARAM.