Can/should I use Jenkins to monitor whether a website is up and send out an email if it's down?
It's possible and pretty simple to do this with Jenkins. Here is a sample pipeline:

pipeline {
    agent any

    triggers {
        cron('*/5 * * * *')
    }

    stages {
        stage('Health checker') {
            steps {
                sh 'curl https://mywebsite.com'
            }

            post {
                failure {
                    mail to: 'notify-list@example.com',
                        from: 'jenkins@example.com',
                        subject: "${env.JOB_NAME} - Failed",
                        body: "Job Failed"
                }
            }
        }
    }
}