The US developed many and very effective AA missiles. BOMARC was an early example and remained in service for over 20 years.
Redeye, Stinger, Hawk, Nike, Patriot, THAAD, Terrier, Talos, Tartar, Standard, Spartan, Sprint, the list goes on and on.
And that's just SAM systems, doesn't include AAM systems, of which several (Sparrow, AMRAAM, ASRAAM, Sidewinder) have found use as SAMs as well.
Of course the number of different systems doesn't come close to the number of different systems employed by the USSR at one time or another, but that's in no small part because of different design and numbering philosophies. When the Soviets designed a new system they usually started from scratch, the Americans tended to do more gradual changes, morphing one system into another over time.
Prime example of this is the US Navy Standard system, now in its 3rd major incarnation but still outwardly similar and using the same name. And it can be argued that Standard is merely a further development of Talos and Terrier. The Nike family is also comprised of several missiles bearing the same name. And at current THAAD is several weapons, Patriot is now in its 3rd incarnation as well.
On the AAM side, AMRAAM is in its 3rd incarnation (though the AIM-120B was not produced), Sidewinder has seen over a dozen variants (including several SAM and ground attack versions, and anti-ship and anti-radar missiles).
Also, they may appear cheaper, but are they? A single missile costing a few million dollars can indeed bring down an aircraft costing a hundred million, but you will likely need several missiles to bring down that aircraft.
Also, missiles are far less flexible than are aircraft, taking longer to forward deploy (especially the larger ones that would replace long range fighters) and are more prone to suffering from environmental problems that aircraft do not. Just ask the British about the trouble with their SAMs during the Falklands campaign.