Public is not the same as basic - there is no "basical" and there is no record of it in the past.
Public was originally an 8th century Anglo Norman adjective[1] that also had a rare form, which appeared around the 14th/15th century "publical"[2]. This would indicate the use of the noun around the same time.
As a noun, it was then recorded in the adverbial phrase "in public"[3] in the early 15th century and increased in popularity as a pure noun, albeit not in the current sense, to the 16th century[4].
It took until the 17th century for the noun "public" to reach its current meaning[5] And all that time there was, lurking in the background, people who used "publical" as an adjective as they had done in the 14/15th century, something that was continued at least until the 19th century[6].
Thus the adjective "public/publical" was possible and its adverb was thus formed with the standard "-ly" suffix. As the difference between the spoken "-icly" and "-ically" is negligible, and as by the 16th century standard spelling was not established, both forms must have existed.
It should be remembered that the reason that any word is spelled the way that it is "because that is how we do it" - there is no authority, merely tradition, and the transition to a completely regular orthography has not yet been completed.
From OED:
[1]Etymology: < Anglo-Norman publik, pupplik, Anglo-Norman and Middle French public, publiq, publique, Middle French publice, publicque, puplique, French public, †publique (adjective) of or relating to the people as a whole (first half of the 13th cent. in Old French),
1394 in Collectanea Topographica & Genealogica (1836) III. 257 (MED) We..to this oure writyn certi[fi]cat han frelich sett oure selys, in ful and pupplik declaracion of trowthe in dede.
[2]Etymology: < post-classical Latin publicalis (8th cent. in a British source; < classical Latin publicus public adj. + -ālis -al suffix1.]
c1450 Alphabet of Tales (1904) I. 248 In gude felowshupp..þer suld all publicall honor and wurshup sese betwix þe fadur & þe son.
[3]B. n.
1. in public.
a. In a public place; before spectators or onlookers; publicly, openly, without concealment. Formerly also †in the public (obsolete). †into public: into print; (also) into the public eye; (obsolete).
c1429 Mirour Mans Saluacioune (1986) l. 923 To convers in publike [glossed in comon place] hase sho fledde all hire lif, Halsing solitarie estat and pure contemplatif.
[4] 2.†a. With singular agreement. The community or people as an organized body, the body politic; the nation, the state; the interest or well-being of the community, the common good. Chiefly with the. Obsolete.
a1586 Sir P. Sidney Arcadia (1590) iii. xix. 325 You are a Prince, & a father of people, who ought..to set downe all priuate conceits, in comparison of what for the publike is profitable.
[5]b. With the and singular or plural agreement. The community or people as a whole; the members of the community collectively.
1621 J. Reynolds Triumphs Gods Revenge: 1st Bk. sig. A3 I only represent what they have acted, and give that to the publike, which they obscurely perpetrated in private.
[6]1898 Catholic World May 240 Even the unselfish and the honest patriots feel a glow of pleasure in the doling out of the loaves and fishes, for they believe that success begets success and state and nation will be their next publical advance.
basicallyis correctly, butpublicallyis not. "Basicly" is just bad English. So why does that not apply here?