I feel like the majority of this is fairly subjective, but even then there's plenty to say on that matter.
The rewards are awful, mostly consisting of key fragments that sit unused.
This is a fair argument, but there really isn't much that we can get beyond what we do. For the largest part, up 'til Honor 5, we can't be assured of any specific reward outside of Key Fragments - offering skin shards would likely be taxing (even if limited to the Honor Levels themselves), and most other content, I imagine, wouldn't actually be all that satisfying.
If you've suggestions on what Riot could put on offer for Honor Rewards, I (and probably most people, Riot or otherwise) would definitely like to hear it.
The grind is extremely long, with consistently positive players who play every day still working on reaching H5.
I personally don't exactly consider this a valid argument - Riot has said that it's intended to be a year-long affair, not something where you can just jam out for a month and hit Honor 5. I suppose there could be something said about Honor gain being too slow, but I think enough players have hit Honor Level 5 here in early-to-mid August that it's hard to say that it is too slow.
Like matchmaking, Honors don't depend on you, they depend on other people, making them effectively random.
Yes, and no. But, let's get something out of the way first - this isn't new. Even with the old Honor System, you couldn't just generate Honors by existing, you had to have other people Honor you. In this regard, the current Honor System is no different from the old one.
But, more to my point - yes, it is true that it is effectively random whether or not you get Honored, since it's at the whims of other players, but it also isn't entirely true that you have no control over it. The way you act does affect your likelihood to get Honored by other players. Hell - even if you don't say a word, you can act in very subtle ways that shift people's perception of you.
For example, when I play engage or pick-focused Champions, playmakers like Pyke, Leona, etc., I virtually always make the first move, and I'd reckon that in a majority of those games, I've gotten a Shotcaller honor.
When simple things like that can have an effect on what Honors you receive (if any), it's hard to actually say that Honors are effectively random.
Each player can only Honor one teammate, making it a popularity contest.
And this is where I'd say it starts getting subjective. I don't intend to insult here, but it feels like points 4 and 5 are personal gripes with other players more than actual issues with the Honor System.
But, regardless, Riot made it so that players can only Honor one teammate because they wanted that Honor to feel more impactful on both ends. They wanted the player Honoring to actually think about who they were going to Honor, about who they felt had the most positive impact on their game, and they wanted the Honored player to feel rewarded when they get Honored.
I don't really understand what you mean by "making it a popularity contest", though. I could (roughly) understand if the argument was "most players honor the carry" or whatever, but, in a game where you're largely matched with players you don't know, I'm having a reasonable amount of trouble trying to understand a popularity contest in this context.
It's possible to Honor someone when no one intended on Honoring you, like holding the door open for someone only to have them cut in line.
Again, this is probably more a gripe with players than with the system. More than that, it seems to be a personal gripe. Why take offense if/when nobody Honors you? You're not guaranteed by any stretch to be Honored every game, and people really aren't under any obligation to Honor you - or anyone.
Honoring other players should not come with the expectation that you will be Honored in kind. That turns it from "Honor who you felt had the most positive impact on the game" to what basically amounts to Honor trading. And at that point, it's not about good behavior, it's just butt-buddy points, for lack of better words.
If I hold the door open for someone, I only expect them to be mildly thankful for the convenience. I don't expect them to say "thanks" (though people often do), I don't expect them to give me a nod or anything. The thanks isn't the point of the action. The point of the action is to benefit someone else.
Likewise, you don't Honor someone to be Honored back. You Honor them to show them that they made the game better for you. You can expect thankfulness, but you shouldn't always expect acknowledgement or payback. That isn't the point of the action.
Instead of good behavior being its own reward, the system encourages people to hope for or even expect an extrinsic reward, and not getting an Honor feels worse than not having an Honor system.
The latter half of this is subjective, and like with point 5, it seems to be more a personal issue on your end with the expectation of being Honored. That's something for you to consider and resolve yourself.
But, for the former half...That, I can at least agree is something of a problem. I think most people can say it's a damn shame when good behavior has to be encouraged through an extrinsic motivator (skins, content, etc.) instead of an intrinsic one. That the Honor system partially serves that end is definitely something of a bad thing.
But. In the same turn, the Honor system also works to reward those who do show positive, sportsmanlike behavior. People who don't need a push to show positive behavior get their behavior recognized through the rewards they get. One could still argue that they still have the expectation of a reward, but...Is it still a problem if the reward is more a foregone conclusion than a motivator?