Summoner icons are awarded as achievements, but there is NO WAY to show off those achievements.
This is the biggest problem with summoner icons at the moment, and the reason they feel bad as rewards. The only way you can show other people your summoner icon is to equip it, and then it's just a cool looking square that shows up next to your name.
Did you get that icon for completing a tough challenge mission in a rotating game mode? Did you get it for being honorable? Is it special? Who knows!
Unless you're someone who goes through a lot of trouble to identify which handful of the many summoner icons signifies what certain pains players went through to unlock them, the merits behind unlocking an icon are a total mystery, which completely defeats the purpose of using them to show off your accomplishments in game.
Unless someone makes an effort to learn all of the specially awarded summoner icons and how they were obtained there's not any difference between special achievement icons and purchased/randomly rolled icons. All these worthwhile achievements, such as completing difficult challenges in Star Guardian Invasion, being among the top honorable players before the honor rework, being a veteran player who has stuck with the game for years, will be completely forgotten within a couple months after the initial announcement and release of the icon.
Do you remember how players were able to unlock these Dark Star summoner icons?
https://am-a.akamaihd.net/image?f=https://news-a.akamaihd.net/public/images/pages/2017/april/dark-star/img/corruption-icon.jpg https://am-a.akamaihd.net/image?f=https://news-a.akamaihd.net/public/images/pages/2017/april/dark-star/img/cosmic-ruins-icon.jpg
...because if you're like me, you had to go look it up right when I asked.
There NEEDS to be some way to display summoner icons on your profile in a way that lets people know what you did to earn them. A row of summoner icons that you can display as badges to show off your accomplishments in game would go a long way to making summoner icons feel worthwhile.
TL;DR: Riot can't award summoner icons as achievements, special merits, and unique rewards UNLESS they add a way to display and share those achievements with others, because unless you actually equip an icon, and somebody you play with happens to know what that icon was earned for, they're completely meaningless besides being just another cool looking square that shows up next to your name.