Should it be the worker's duty to constantly be searching for better opportunities and interviewing for new jobs while working - or the employer's duty to provide reasonable CoL raises so employees aren't driven away from the company?
I don't think either side owes the other anything. But that goes both ways and it can mean losing half your staff in a year for a business and training workers who have no loyalty to your business.
Everyone should be operating in their own best interest. If it's in the company's best interest to raise your pay to keep you on, they will. If it's in your best interest to leave, you should.
Some of my favorite career advice is to seek out a minimum of one interview per year. This lets you see what skills are in demand, what salaries are being offered, and practice your interviewing skills. It's a great way to educate yourself and improve long term career growth.
"Should" being the key word here. Companies are no more rational than its individual humans, which is why they are often irrationally stingy to the point of massive costs.
The only "duty" your employer has is to extract as much value out of you for as cheap as possible. If you expect anything different then you are going to be taken for a ride.
An employer at a for-profit company is responsible for maximizing the company's profits, but that doesn't mean it has a duty to extract as much value out of an employee for as cheap as possible.
Such an adversarial relationship with employees causes them to leave, or at least avoid initiative or innovative solutions that are mutually beneficial to the employer and employee.