It's actually not required that you must choose a sweepstakes winner at random, but that's because sweepstakes is a broad term which may apply to non-chance contests or not, depending on the law and definition. In Washington under RCW, there is a distinction between a promotion and a promotional contest of chance where the former is "an advertising program, sweepstakes, contest, direct giveaway, or solicitation directed to specific named individuals, that includes the award of or chance to be awarded a prize" but excludes "a promotional contest of chance under RCW 9.46.0356(1)(b). What you want to avoid is being found to be running a gambling enterprise, because that is a crime if you don't have a license. To be convicted of gambling, the state (not you) must prove that your activity is gambling, which means proving that it is not "a contest of chance which is specifically excluded from the definition of lottery under this chapter shall not constitute gambling". The statute that authorizes promotional contests of chance says "Promotional contests of chance under this section are not gambling as defined in RCW 9.46.0237". In other words, the burden of proof rests on the state to prove beyond reasonable doubt that your activity is not a promotional contest of chance.
All the law says about "contest of chance" is the definition: "any contest, game, gaming scheme, or gaming device in which the outcome depends in a material degree upon an element of chance, notwithstanding that skill of the contestants may also be a factor therein". There is no monitoring, which would be extremely burdensome.