This should have been discussed with the landlord as it happened. What really should have happened is that the landlord should have provided new appliances as the old ones wore out.

Or the landlord should have made it clear at the start of the lease that the appliances were the tenant's responsibility.

What I'd do as the tenant in the situation in the OP is contact the landlord and ask if he/she wants to buy the appliances. If the landlord doesn't want to buy them, then the tenant gets to take them when they leave.

There's no way the tenant has to buy new appliances. And if the tenant is moving somewhere where he doesn't want to take the appliances, just ask the landlord if it is okay if they are left behind. Most likely the landlord would rather get free appliances than have to go out and buy new ones.

Google & read your state landlord/tenant law. In my state if appliances are provided by LL/owner as an inducement to rent their property, then LL must maintain them, or replace if need arises. You might have similar laws in your state.