If someone is caring for an elderly parent in that parents house, but are living next door, and they are left the house, that would be considered unearned income I imagine.
In a technical sense it is, yes. Not sure how you'd design a tax system to provide greater credits to people who spend more time and effort in caring responsibilities, from a practical POV. There is an exemption for people who live with their parent and inherit the house.
It is more a definition thing. Calling one type "unearned" is mostly to be insulting/downgrade that type of income. I think labour income and investment income would be better names.
The earned/unearned distinction doesn't really stand up to scrutiny. Deciding how to allocate capital is work.
The “earned vs unearned” distinction is a standard technical classification used in tax systems. For example, the Internal Revenue Service explicitly defines those categories using that wording. Other tax systems work on the same basic technical basis: it’s just shorthand for income from labour vs income from capital. Even if investing involves effort, the income itself is treated as arising from ownership of assets rather than ongoing labour - that’s the distinction the system is making.
You could absolutely call them “labour income” and “investment income” instead, but that’s basically just a different label for the same underlying classification that most tax systems rely on
I think a reasonable compromise would be to remove the CGT exemption on death.
The exit tax approach seems fair: death is a taxable event, but the exit tax paid is available as a CAT credit to the recipient.
From a technical standpoint this would materially change the tax base and distribution of the burden, and introduce significant administrative complexity, particularly if the intent is to reduce the burden on the recipient.
The current CAT regime is relatively transparent and predictable (e.g. child with a defined threshold, tax on value above that). Moving to a CGT charge on death would instead require all assets to be valued and historic base costs established, which is often impractical in practice.
More fundamentally, it shifts the system from taxing inheritances (value received) to taxing historic gains, which can produce highly uneven outcomes. For example, long-held quoted shares with low base cost would trigger substantial CGT, whereas a principal private residence of equivalent value would (presumably) remain exempt. Two estates of similar value could therefore face very different effective tax rates depending purely on asset composition and holding period.
It would also create behavioral distortions around timing of disposals and estate planning, and raise liquidity issues where tax becomes payable on death without a corresponding realization event.
I think tinkering around the edges would make sense - I have no real issue with favoured nieces and nephews for people without children, for example, so long as it's within reasonable limits.
I did look it up in more detail, and indeed organized day trading is classified as earned income. Passive investments as unearned income. Taking however a purely personal point of view I struggle with the definition that rental income is "unearned", even if that is the tax classification. Being a landlord can involve plenty of work, just not a regular every-day all-day work. Just recently I had to drop what I was doing and urgently deal with a leak! 
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