Is there an alternative to having the HR department responsible for getting employee reviews done?
Yes. It's called "cascading responsibility". Since each manager is accountable to his or her own manager, and NOT to HR, the alternative is simple. At the top the CEO is responsible for evaluating his or her reports, and includes an objective/responsibility for them to ensure that THEIR reports do their appraisals, and manage performance properly and in a timely manner. It cascades down.
This is pretty much the only practical enterprise wide way to make performance management work.
It's fine for HR to be involved in recording keeping, and housing the paperwork, or computer reports associated with performance appraisal. It's NOT fine for them to operate as "nags" to get them done, since they lack the line authority over managers.