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I thought it did, and I wanted to test it to prove it. I used Riot's API to query the last 100 matches of a summoner and then go through them all and count the champs that appear in ARAM games. I ran a couple of my friends and I'll include the results here. You may not believe the numbers, I didn't either - so I ran my last 20 games and compared the results by hand. The numbers were correct.
The data is laid out as Champ: # of appearances, overall %
Friend 1
Aatrox: 3, 3%
Ahri: 11, 13%
Akali: 16, 19%
Alistar: 3, 3%
Amumu: 2, 2%
Anivia: 3, 3%
Annie: 10, 12%
Aphelios: 3, 3%
Ashe: 10, 12%
AurelionSol: 3, 3%
Azir: 7, 8%
Bard: 4, 4%
Blitzcrank: 13, 16%
Brand: 11, 13%
Braum: 2, 2%
Caitlyn: 14, 17%
Camille: 1, 1%
Cassiopeia: 4, 4%
Chogath: 8, 9%
Corki: 6, 7%
Darius: 8, 9%
Diana: 6, 7%
DrMundo: 4, 4%
Draven: 5, 6%
Ekko: 6, 7%
Elise: 5, 6%
Evelynn: 8, 9%
Ezreal: 11, 13%
Fiddlesticks: 5, 6%
Fiora: 0, 0%
Fizz: 5, 6%
Galio: 2, 2%
Gangplank: 5, 6%
Garen: 10, 12%
Gnar: 2, 2%
Gragas: 4, 4%
Graves: 11, 13%
Hecarim: 2, 2%
Heimerdinger: 9, 11%
Illaoi: 6, 7%
Irelia: 5, 6%
Ivern: 1, 1%
Janna: 3, 3%
JarvanIV: 3, 3%
Jax: 1, 1%
Jayce: 4, 4%
Jhin: 12, 14%
Jinx: 5, 6%
Kaisa: 13, 16%
Kalista: 2, 2%
Karma: 4, 4%
Karthus: 4, 4%
Kassadin: 3, 3%
Katarina: 4, 4%
Kayle: 8, 9%
Kayn: 3, 3%
Kennen: 9, 11%
Khazix: 3, 3%
Kindred: 11, 13%
Kled: 2, 2%
KogMaw: 3, 3%
Leblanc: 7, 8%
LeeSin: 7, 8%
Leona: 7, 8%
Lissandra: 3, 3%
Lucian: 9, 11%
Lulu: 3, 3%
Lux: 13, 16%
Malphite: 9, 11%
Malzahar: 5, 6%
Maokai: 1, 1%
MasterYi: 5, 6%
MissFortune: 13, 16%
MonkeyKing: 2, 2%
Mordekaiser: 10, 12%
Morgana: 10, 12%
Nami: 8, 9%
Nasus: 11, 13%
Nautilus: 7, 8%
Neeko: 14, 17%
Nidalee: 5, 6%
Nocturne: 2, 2%
Nunu: 4, 4%
Olaf: 3, 3%
Orianna: 5, 6%
Ornn: 6, 7%
Pantheon: 5, 6%
Poppy: 3, 3%
Pyke: 3, 3%
Qiyana: 3, 3%
Quinn: 3, 3%
Rakan: 2, 2%
Rammus: 0, 0%
RekSai: 0, 0%
Renekton: 4, 4%
Rengar: 4, 4%
Riven: 3, 3%
Rumble: 3, 3%
Ryze: 7, 8%
Sejuani: 0, 0%
Senna: 5, 6%
Shaco: 4, 4%
Shen: 5, 6%
Shyvana: 4, 4%
Singed: 3, 3%
Sion: 1, 1%
Sivir: 14, 17%
Skarner: 2, 2%
Sona: 7, 8%
Soraka: 9, 11%
Swain: 7, 8%
Sylas: 5, 6%
Syndra: 3, 3%
TahmKench: 3, 3%
Taliyah: 7, 8%
Talon: 4, 4%
Taric: 2, 2%
Teemo: 9, 11%
Thresh: 12, 14%
Tristana: 6, 7%
Trundle: 1, 1%
Tryndamere: 2, 2%
TwistedFate: 5, 6%
Twitch: 5, 6%
Udyr: 0, 0%
Urgot: 5, 6%
Varus: 8, 9%
Vayne: 8, 9%
Veigar: 8, 9%
Velkoz: 9, 11%
Vi: 2, 2%
Viktor: 7, 8%
Vladimir: 5, 6%
Volibear: 1, 1%
Warwick: 9, 11%
Xayah: 5, 6%
Xerath: 4, 4%
XinZhao: 2, 2%
Yasuo: 3, 3%
Yorick: 6, 7%
Yuumi: 5, 6%
Zac: 1, 1%
Zed: 9, 11%
Ziggs: 12, 14%
Zilean: 6, 7%
Zoe: 7, 8%
Zyra: 3, 3%
ARAM matches played: 81
Friend 2
Aatrox: 2, 6%
Ahri: 8, 25%
Akali: 2, 6%
Alistar: 1, 3%
Amumu: 2, 6%
Anivia: 1, 3%
Annie: 4, 12%
Aphelios: 0, 0%
Ashe: 4, 12%
AurelionSol: 0, 0%
Azir: 3, 9%
Bard: 0, 0%
Blitzcrank: 2, 6%
Brand: 5, 15%
Braum: 0, 0%
Caitlyn: 2, 6%
Camille: 2, 6%
Cassiopeia: 2, 6%
Chogath: 2, 6%
Corki: 3, 9%
Darius: 3, 9%
Diana: 2, 6%
DrMundo: 2, 6%
Draven: 0, 0%
Ekko: 2, 6%
Elise: 1, 3%
Evelynn: 2, 6%
Ezreal: 4, 12%
Fiddlesticks: 0, 0%
Fiora: 0, 0%
Fizz: 1, 3%
Galio: 5, 15%
Gangplank: 1, 3%
Garen: 3, 9%
Gnar: 0, 0%
Gragas: 1, 3%
Graves: 4, 12%
Hecarim: 1, 3%
Heimerdinger: 4, 12%
Illaoi: 1, 3%
Irelia: 3, 9%
Ivern: 0, 0%
Janna: 5, 15%
JarvanIV: 1, 3%
Jax: 1, 3%
Jayce: 1, 3%
Jhin: 3, 9%
Jinx: 5, 15%
Kaisa: 1, 3%
Kalista: 3, 9%
Karma: 4, 12%
Karthus: 3, 9%
Kassadin: 0, 0%
Katarina: 3, 9%
Kayle: 5, 15%
Kayn: 6, 18%
Kennen: 0, 0%
Khazix: 1, 3%
Kindred: 3, 9%
Kled: 1, 3%
KogMaw: 0, 0%
Leblanc: 5, 15%
LeeSin: 4, 12%
Leona: 1, 3%
Lissandra: 3, 9%
Lucian: 2, 6%
Lulu: 1, 3%
Lux: 6, 18%
Malphite: 5, 15%
Malzahar: 1, 3%
Maokai: 4, 12%
MasterYi: 6, 18%
MissFortune: 2, 6%
MonkeyKing: 2, 6%
Mordekaiser: 2, 6%
Morgana: 6, 18%
Nami: 2, 6%
Nasus: 4, 12%
Nautilus: 1, 3%
Neeko: 2, 6%
Nidalee: 5, 15%
Nocturne: 0, 0%
Nunu: 1, 3%
Olaf: 1, 3%
Orianna: 3, 9%
Ornn: 2, 6%
Pantheon: 0, 0%
Poppy: 2, 6%
Pyke: 1, 3%
Qiyana: 0, 0%
Quinn: 3, 9%
Rakan: 1, 3%
Rammus: 1, 3%
RekSai: 0, 0%
Renekton: 1, 3%
Rengar: 2, 6%
Riven: 2, 6%
Rumble: 2, 6%
Ryze: 3, 9%
Sejuani: 1, 3%
Senna: 2, 6%
Shaco: 2, 6%
Shen: 2, 6%
Shyvana: 1, 3%
Singed: 3, 9%
Sion: 1, 3%
Sivir: 0, 0%
Skarner: 0, 0%
Sona: 3, 9%
Soraka: 2, 6%
Swain: 1, 3%
Sylas: 3, 9%
Syndra: 2, 6%
TahmKench: 0, 0%
Taliyah: 3, 9%
Talon: 0, 0%
Taric: 0, 0%
Teemo: 3, 9%
Thresh: 4, 12%
Tristana: 4, 12%
Trundle: 0, 0%
Tryndamere: 2, 6%
TwistedFate: 3, 9%
Twitch: 1, 3%
Udyr: 2, 6%
Urgot: 3, 9%
Varus: 4, 12%
Vayne: 1, 3%
Veigar: 6, 18%
Velkoz: 0, 0%
Vi: 0, 0%
Viktor: 3, 9%
Vladimir: 3, 9%
Volibear: 1, 3%
Warwick: 2, 6%
Xayah: 4, 12%
Xerath: 3, 9%
XinZhao: 2, 6%
Yasuo: 2, 6%
Yorick: 2, 6%
Yuumi: 5, 15%
Zac: 1, 3%
Zed: 2, 6%
Ziggs: 1, 3%
Zilean: 2, 6%
Zoe: 5, 15%
Zyra: 3, 9%
ARAM matches played: 32
Friend 3
Aatrox: 10, 10%
Ahri: 12, 12%
Akali: 10, 10%
Alistar: 4, 4%
Amumu: 2, 2%
Anivia: 4, 4%
Annie: 9, 9%
Aphelios: 8, 8%
Ashe: 15, 15%
AurelionSol: 5, 5%
Azir: 8, 8%
Bard: 4, 4%
Blitzcrank: 12, 12%
Brand: 9, 9%
Braum: 1, 1%
Caitlyn: 15, 15%
Camille: 1, 1%
Cassiopeia: 4, 4%
Chogath: 9, 9%
Corki: 8, 8%
Darius: 8, 8%
Diana: 4, 4%
DrMundo: 8, 8%
Draven: 5, 5%
Ekko: 8, 8%
Elise: 1, 1%
Evelynn: 4, 4%
Ezreal: 15, 15%
Fiddlesticks: 7, 7%
Fiora: 3, 3%
Fizz: 6, 6%
Galio: 4, 4%
Gangplank: 8, 8%
Garen: 9, 9%
Gnar: 5, 5%
Gragas: 2, 2%
Graves: 8, 8%
Hecarim: 4, 4%
Heimerdinger: 8, 8%
Illaoi: 7, 7%
Irelia: 5, 5%
Ivern: 4, 4%
Janna: 6, 6%
JarvanIV: 6, 6%
Jax: 6, 6%
Jayce: 4, 4%
Jhin: 8, 8%
Jinx: 7, 7%
Kaisa: 14, 14%
Kalista: 1, 1%
Karma: 13, 13%
Karthus: 3, 3%
Kassadin: 3, 3%
Katarina: 6, 6%
Kayle: 8, 8%
Kayn: 1, 1%
Kennen: 3, 3%
Khazix: 2, 2%
Kindred: 12, 12%
Kled: 1, 1%
KogMaw: 6, 6%
Leblanc: 9, 9%
LeeSin: 7, 7%
Leona: 11, 11%
Lissandra: 3, 3%
Lucian: 7, 7%
Lulu: 3, 3%
Lux: 8, 8%
Malphite: 3, 3%
Malzahar: 9, 9%
Maokai: 6, 6%
MasterYi: 9, 9%
MissFortune: 9, 9%
MonkeyKing: 5, 5%
Mordekaiser: 6, 6%
Morgana: 19, 19%
Nami: 8, 8%
Nasus: 12, 12%
Nautilus: 9, 9%
Neeko: 12, 12%
Nidalee: 7, 7%
Nocturne: 4, 4%
Nunu: 4, 4%
Olaf: 1, 1%
Orianna: 4, 4%
Ornn: 8, 8%
Pantheon: 4, 4%
Poppy: 5, 5%
Pyke: 9, 9%
Qiyana: 3, 3%
Quinn: 6, 6%
Rakan: 5, 5%
Rammus: 2, 2%
RekSai: 3, 3%
Renekton: 7, 7%
Rengar: 2, 2%
Riven: 2, 2%
Rumble: 4, 4%
Ryze: 6, 6%
Sejuani: 1, 1%
Senna: 6, 6%
Shaco: 5, 5%
Shen: 10, 10%
Shyvana: 5, 5%
Singed: 3, 3%
Sion: 4, 4%
Sivir: 12, 12%
Skarner: 6, 6%
Sona: 4, 4%
Soraka: 17, 17%
Swain: 11, 11%
Sylas: 12, 12%
Syndra: 5, 5%
TahmKench: 5, 5%
Taliyah: 4, 4%
Talon: 3, 3%
Taric: 3, 3%
Teemo: 12, 12%
Thresh: 12, 12%
Tristana: 7, 7%
Trundle: 1, 1%
Tryndamere: 4, 4%
TwistedFate: 4, 4%
Twitch: 9, 9%
Udyr: 2, 2%
Urgot: 8, 8%
Varus: 12, 12%
Vayne: 9, 9%
Veigar: 12, 12%
Velkoz: 8, 8%
Vi: 2, 2%
Viktor: 2, 2%
Vladimir: 10, 10%
Volibear: 0, 0%
Warwick: 7, 7%
Xayah: 16, 16%
Xerath: 3, 3%
XinZhao: 2, 2%
Yasuo: 8, 8%
Yorick: 6, 6%
Yuumi: 10, 10%
Zac: 1, 1%
Zed: 9, 9%
Ziggs: 8, 8%
Zilean: 5, 5%
Zoe: 11, 11%
Zyra: 10, 10%
ARAM matches played: 96