A little more effort may be required than that. You can expect that when and if the original Hawaii records turn up, there will be some kind of persuasive record of a Hawaii birth--maybe home birth; maybe something else. The reason the campaign is concealing those records is because there is evidence in them that would lead to the real story if pursued.
The most helpful evidence that could be obtained is two things: A birth record from Kenya demonstrating that he was born there; and, a marriage record from Kenya demonstrating that Stanley Ann and Obama Sr. were married there. The birth record search should be from mid July to early August, 1961; the marriage record search should be from January 1961 to April 30, 1961.
Other things should be done--the address in the birth announcement should be searched at the title company as well as should the neighboring properties; people who owned, leased, or occupied at that time period should be contacted.
As everyone will recall, Kenya was a British colony in this period and adequate records will have been kept. The suggestion has appeared that those records were moved to London with copies remaining in Mombasa or Nairobi.
Fiche copies are reportedly located in Salt Lake City with a Historical Genealogy Library. The Mormon birth records may also contain copies. Anyone in that area who could do so should undertake a search.