How does the responsibility for executing the law give the president great power?

Answer 1

As the person ultimately responsible for the enforcement of laws, the President has considerably power in determining three major factors:

1. The priority and resource allocation of various law enforcement divisions - as there is never enough time, personnel and money to enforce all laws equally, the President has the ability to stipulate that certain laws be given more attention to enforcement. This can result in the appearance that certain laws aren't being followed; however, this merely reflects the position that certain laws are less "worthy" of having limited resources spent on them. You can still be arrested and convicted of those "neglected" laws, but the likelihood of this happening is very small, as law enforcement is simply not paying any attention to violators of those laws.
2. How any laws are defended against legal challenges - the President is charged with the defense of all laws in the court system. Like the above, the President can set priorities for how much a law is to be defended, and can also direct the particular legal emphasis by which to defend the law, which can have a very significant effect on the ultimate judicial interpretation.
3. As all laws are inherently vague, the President can set the specific interpretation of that law as to it's enforcement. That is, the President can say "This law means X is illegal". Ultimately, the courts are responsible for deciding if this interpretation is correct, but the President has an enormous influence in setting this original interpretation.

The President (and Executive Branch) has no ability to invalidate a law, or declare it unConstitutional, or otherwise fail to enforce a valid law. The first two powers are strictly the responsibility of the Judicial Branch. In the latter case, citizens can sue the Executive to force enforcement if the Executive decides to "negate through neglect" by not allocating law enforcement resources.

The Executive can also decide that certain laws are unConstitutional and stop trying to defend them in any pending judicial court case. This is NOT the same as failing to enforce such laws - the Executive is bound to continue to enforce that law, up until a judicial case declares the law null and void. However, in that court case, the Executive branch (as the nominal "defense") can simply say "We agree with the plaintiff's case that the law should be overturned". The Judicial branch is still the ultimate decider of whether the law is Constitutional or not, but having the defense effectively agree with the plaintiff is a very strong motivator (but NOT a requirement) for the Judicial branch to overturn the law.

Answer 2

The responsibility for executing the new law would give a president great power because he has final say. The president could take out an officer that was doing bad or wrong and replace him immediately with someone worthy of doing the job.