A. The president has grown tired of waiting. This is the correct answer.
The underlined verb is linking or connecting the two most important pieces of the sentence: The president and tired of waiting. The predicate refers to the physical or emotional state of the subject; the president has grown / got / become tired of waiting. These verbs just reflect the process or the change of mood of the president. The president was not tired. Something happended and made him be tired. Therefore, the president has grown tired of waiting.
These other options do not have a linking verb.
B. The general shall lead us to victory. ("To lead" is an action verb. It refers to the action of the subject - the general- and the effect this action has on another person-us-.)
C. The senate has looked at the results of the vote. ( " To look" is a sense verb . The subject - the senate- has decided to look at something : the results of the votes).
D. The senate should have debated the new law. ( " To debate" shows a verbal action. There is a speaker - the senate- and a topic : the new law).