Third party security system does not come without price, which sometime is hefty and unbearable and may be against your national interests and aspirations.

US defense system in Qatar too has the same implications like,

going after Libya,
going after Syria
going after Iraq
going after Afghanistan

All these countries have no common border with Qatar, and no direct border or business dispute, yet Qatar's soil was used to make them all unstable lands, and as a result, created an unavoidable bigger threat for the regional security, which a challenge is now all around the Gulf.