Actual decline vs. learned helplessness...

I can't say for sure because I am not qualified to make the diagnosis, but my mom also has all of these behaviors and was diagnosed with dementia a couple of years ago. Like you I am often confused whether this is just helplessness or the disease. In the past two years since she was diagnosed, I have come to see these behaviors as progression of the disease. It's not that she doesn't want to do these things, it is that she is becoming increasingly less capable of doing these things.

Does your elderly relative live alone? They may very well have the beginnings of dementia or even a very mild case. When you say the sundowning is very prominent, what happens when it starts? What do they do?
I ask this because my mother (84) when it starts getting dark often starts working herself up into a panic attack. Rocking back and forth with her hands over her face like she's crying.
I completely ignore her of course because this has been going on since she was 40 years old.
It's a performance with her. Often elderly people give performances the same as a child when they're not getting the amount of attention they want.
Learned helplessness is a real thing. Her caregiver has to make her do chores that she is able to do. Like folding her own laundry and putting it away. The caregiver may have to assist, but make her do it.
As for not carrying dishes to the kitchen. There's no reason why she should be taking her meals anywhere besides the kitchen or a dining room if there is one. Instruct the caregiver to prepare her meal then call her into the kitchen to eat at the table. Never allow meals to be taken all over the house because it will become a habit.
Please instruct the caregiver on assisting your elderly relative with ADL's like reading the mail. Or putting the laundry away. Or properly wrapping up food to store it.
If your relative lives alone and the overly dependent behavior continues or gets worse, then they can't live alone anymore. They will need a live-in caregiver or placement in an assisted living facility.

This sounds like actual decline in every instance and especially in those areas where this is CHANGE from this person's norm. The purpose of OT and PT personnel in any rehab is to find ways to engage and to guide a person in self care. However I am interested in the fact that you are not mentioning short term memory issues.
Without participation on those who are well, the person in decline will likely continue in this trajectory even to the point of all hygiene, the ability to dress oneself and etc. It requires the same sort of constant guidance much in the way that a toddler does to keep the person motivated, and with some dementias there is no motivation, and in fact pushback. Certainly executive functions such as mail and bills fall away very very quickly in many cases. It is time now for an evaluation that is complete and involves the neuro-psychiatric branch as well as some guidance for caregivers in maintaining as much self caring as possible.

I think that's really a difficult diagnosis to make.   And w/o being facetious, some people exhibit those symptoms and are nowhere near to having dementia.  They're just lazy, or don't care about their surroundings or lifestyle.

I think sundowning is a good indicator that changes have taken place though.   

You ask about learned helplessness; did this exist before your relative segued into these behaviors?   

You stated that this relative clearly has dementia but hasn't been diagnosed.   Is there some reason why this person hasn't seen a doctor for specific determination of dementia?