The construction of 惊 follows the essence of the phono-semantic compound character. In folk culture, people who are not well-educated often invent characters in this way. Some of these characters became popular and then were included in the officially admitted simplified forms, and the others were eliminated. For example
(admitted formal simplified forms) 竊 -> 窃, 響 -> 响, 癢 -> 痒, 憂 -> 忧, 認 -> 认
(obsolete forms) 原 -> , 菜 -> , 灌 -> , 穩 -> , 酒 ->
You may have found some regular pattern, when such a character is invented, they tended to use the simplest character as the phonetic part. That is the reason for why not 憼 or 𢢩 --
京 is simpler and well-known.
For another question why 駭 as a synonym of 驚 was simplified just as 骇： though 駭 is a synonym of 驚, it is (and was) not often used by the under-educated people, so it wasn't invented by them and then simplified following another rule independently.
The history of the simplified 惊 is very interesting. My conclusion is:
惊 as the simplified form of 驚, was invented between the end of Qing Dynasty and the first decades of the Republic of China (i.e. around the beginning of the 20th century), and formally seen in 手頭字第一期字彙. Interestingly, it happened to be the same as a variant form of 悢.
The relationship among 驚, 悢, and 惊 can be illustrated in the figure below (full size):
1. How and When did 惊 become the simplified form of 驚
汉字字源: 当代新说文解字 by 窦文宇 and 窦勇 says
Seal character and modern traditional character 驚 is composed of 敬 and 马. "敬" has a meaning "let people be obedient", and the whole character means "a horse isn't obedient", then extends in meaning "a horse goes mad", and then "startle, scare, surprise, etc." The simplification follows the folk character form in modern times. The simplified 惊 is a phono-semantic compound character: 心 is the semantic part and 京 is the phonetic part.
And a better book -- which points out a good reference 手头字第一期字汇(手頭字第一期字彙) --
简化字溯源 by 张书岩, 王铁昆, 李青梅 and 安宁 says
【惊】The simplified character of "驚". "惊" was created by the masses in modern times. It was first seen in
Annotation: 手头字 is an old name for simplified characters, and 字汇 is a collection of characters.
The main points this book states, are
1) 惊 as the simplified form of 驚 was created following the construction method of phono-semantic compound characters;
2) 惊 as the simplified form of 驚 was first formally seen in 手頭字第一期字彙(1935).
This is the mainstream view (As already pointed out in @Phil's answer). But, are they reliable?
For Point 1), as I have analyzed in the "Brief Answer" above, 惊 can be constructed in that way, but it doesn't have to -- the character 惊 had appeared for a long time as a variant character of 悢. So there's possibility that 惊 was borrowed to be the simplified form of 驚 (such a case isn't rare, like 葉->叶). However, because 惊 wasn't simplified by just an individual, it's difficult to find out the original motivation. Anyway, I personally accept Point 1), so I said in the Brief Answer they happened to be the same.
For Point 2), it is possible for researchers to prove that, though still difficult, because proving "惊 had not been (once popularly) used as another form of 驚 before" needs much verification work. Personally, I also accept it, and I will demonstrate it in this logic:
惊 was formally stated as a variant character of 悢 in authoritative dictionaries, and in the past, educated people were not so many, thus very probably they did follow the dictionaries well. That is to say, if the newer dictionaries inherited the fact "惊 is a variant character of 悢" well, 惊 was not so possible to be used as 驚 in that times. => So I just have checked some dictionaries from 說文解字(A.D. 100-121) to 康熙字典(A.D. 1716).
2. Inheritance Chain: 惊 as a Variant Character of 悢
We will see, after 集韻, "惊 as a Variant Character of 悢" was well kept in dictionaries. In these one thousand years, people would not use 惊 as the simplified character of 驚.
3. Modern Times: 惊 as the Simplified Character of 驚
I have checked some other resources for folk characters, like
敦煌俗字典(2005) by 黄征, but I couldn't find any clue showing 惊 was used as the simplified character of 驚 before 1900s.
Maybe you can find some text on ctext.org shows
《黃帝內經·經脈》: 是動則病洒洒振寒，善呻，數欠，顏黑，病至則惡人與火，聞木聲則惕然而 惊 ，心欲動，獨閉戶塞牖而處。
《通典·水利田》: 泉流灌 惊 ，所以育五穀也。
《桃花扇·第六回阮學士懷怨進讒 楊知縣登樓報因》： 如此 惊 慌！」龍友說：「今日清議堂議事，阮圓海對著大眾，說你與左寧南侯有舊，常通私書，將為內應。
《白圭志·第二回絕張宏廬山從學 遇菊英月下訂盟》： 歌罷琴息，庭瑞 惊 喜欲狂，暗思：「此必才女所作也。
《廉吏傳·廉吏传》： ...其令长贪洁吏民良猾悉知其状郡内 惊 竦莫不震慑发兵击慈斩之贼既溃平...
However, after checking these classic works with their original scan versions, I just found those characters are
驚 but not
惊. Such mistakes may be made by the software conversion from simplified Chinese to traditional Chinese.
So, very probably the statement that "It was first seen in 手头字第一期字汇" in the book 简化字溯源 is true. Amazingly, it is very very difficult to find a clear copy of 手头字第一期字汇. Currently (before I post this answer :D), the best version one can directly find by Google would be this one:
Hmm, and after very tough search work, finally I found this piece of newspaper:
申報 Feb. 24, 1935. No. 22210. (Shun Pao). It is so valuable that I decide to upload the whole picture here (full size). Please ignore the advertisement on the newspaper.
The table in the middle is the full list of 手頭字第一期字彙. With an introductory article 手頭字之提倡:
發起人丁淑靜·萬迪鶴萬家寶·小默·王人路·豐子愷·方光燾·巴金·王紀 ·王獨清·王特夫·王國秀·王集從·王屏南·方景略·葉聖陶·朱自清·葉放·左胥之·白薇·葉龢士·朱少卿·朱文叔·任白戈·劉延陵·劉廷芳·劉良模·老舍·余之介·沈子丞·李公朴·吳文祺·沈西苓·沈體蘭·沈志遠·徐澤霖·李長之·米星 ·艾思奇·沈兹九·沈起予·李南薌·李冠芳·陸高誼·吳朗西·吳組緗·吳研因·吳清友·蘇雪林·艾寒村·仿聯德·李貽燕·余楠秋·吳敬恒·吳廉銘·沈端先·李輝英·杜鋼百·艾蕪·辛樹幟·汪靜之·吳翰雲·劉薰宇·伯韓·汪馥泉·吳耀宗·邵力子·孟十還·周木齋·周予同·林漢達·林本僑·東平·金兆梓·金仲華·歐陽山·周伯棣·周伯勳·邵宗漢·羅叔和·阿英·邰爽秋·周越然·征農·金燄·胡仲持·胡風·洪深·姚紹華·姜琦·柳湜·范揚·胡愈之·郁達夫·夏丐尊·倪文宙·祝百英·奚如·祝佛朗·馬宗融·草明·唐弢·馬星野·孫俍工·孫師毅·高夢旦·馬國亮·馬國英·席滌塵·高鐵郎·徐蔚南·徐懋庸·郭一岑·章乃器·曹小端·陳子展·張天翼·曹禮吾·陶行知·張仲實·陸衣言·張肖梅·張良輔·陳君冶·陳克承·陳君涵·許幸之·郭沫若·曹亮·陳致道·郭挹清·陳望道·張夢麟· 彬龢·畢雲程·許傑·許達年·許欽文·曹聚仁·陳維姜·陳端志·陸德音·陳樟生·章錫琛·陸錫值·庶謙·張耀翔·黄石·顧君義·傅東華·馮和法·黄素封·舒新城·黄源·程演生·顧樹森·黑嬰·靳以·臧克家·楊青田·楊東蓴·路敏行·葛喬·葛綏成·楊潮·楊騷·楊霽雲·趙義凭·熊昌翼·趙家璧·蒯斯嚑·趙景深·管萃眞·蔡元培·潘公弼·潘式·樊仲雲·鄭君里·鄭伯奇·蔡希陶·魯彦·鄭振鐸·黎烈文·蔣徑三·鄧裕志·樂嗣炳·蔣弼·蔡慕暉·黎錦明·蔣鏡芙·盧冀野·穆藕初·謝六逸·鍾天心·謝扶雅·鍾韶琴·聶紺弩·魏猛克·潭友六·
Now we can find
驚->惊 is actually on the list. Interestingly, on Aug. 21, 1935, 第一批簡體字表 published by the Republic of China Government didn't include this simplified character: