No I don't see the problem? Please try to give an example situation.

Let me sketch the following situation and tell me what would be an issue:
Bitcoin starts to incorporate UTXO state hashes (of the previous block) in new blocks.
I'm a new node and I want to join the bitcoin network. I download an UTXO set of 500 blocks ago, including the blockchain from that point to present (so the chain length is 500 blocks), from a random person I cannot trust blindly.
Because I cannot blindly trust the UTXO state is valid, what I can do to gain trust in the validity: I start verifying the chain of 500 blocks. The chain turns out to be valid.

To gain more trust I can do 2 things:
- Query a lot of nodes in the network if they also have the same UTXO state hash as me in their UTXO-state-hash-history. (trust the network)
- Progressively download an UTXO state+chain from longer ago, and keep verifying forward. This can be done in small steps or big steps, all the way until the genesis block, gaining more and more trust, the older the UTXO state. (verify myself)

What reason would there be not to trust its validity? Please sketch a scenario.

As vjudeu pointed out, you could get a false UTXO state and chain being sent to you (from an altcoin for example), but you can query other nodes to quickly verify if you're dealing with a false UTXO state, to gain trust. You can also progressively download older UTXO states+chain from other nodes, to gain trust. You can also look at the difficulty of the chain, as altcoins have lower mining difficulty.

That is exactly the goal yes, to make blocks mined 10 years ago obsolete to download/verify.
Nodes do not need at least one peer with the full blockchain.

What makes you trust the genesis block of bitcoin, and consequently all blocks after that? It is the amount of PoW (difficulty) and the chain size. Imagine you download the UTXO set of +-3 years ago, with +-100GB of blocks since that moment to present. Why would a chain of 100GB of blocks, having an UTXO hash in every block, be less trustable? The block difficulty is so high. It would be the same as joining bitcoin 3 years after it's launch.

Altcoins are altcoins because they use different algo's. If not, they would just be bitcoin. The attack you describe is something like a 51% attack.

What would stop an attacker currently to create a longer chain with extremely low mining difficulty from distributing that as the 'real' bitcoin?

You are right about the backwards compatibility, so this proposal would indeed not eliminate the need for a full node to exists, it just gives a possibility to gain some trust in your downloaded UTXO without checking the full chain before you can get started.


The blockchain size will reach 500GB soon, requiring new nodes to download and process 500GB before they can trust anything... The number of full nodes is declining as well.

I know this idea is known already, and some small discussions can be found on the mailing list and in previous topics, however I have not seen a strong conclusion that it is not possible or feasible.
I see AssumeUTXO is an experiment trying to achieve the same thing, also looking to MAYBE include UTXO hashes in blocks in the far future....

If in every block, an hash of the UTXO-Set / UTXO-state (after applying the TX's of the previous block) is included, this would eliminate the NEED for full-nodes (and an initial block download for new nodes). This way, only the latest N blocks and latest N UTXO-set needs to be stored & distributed among peers.

Most nodes can store the UTXO data, which does not grow rapidly. Verification of a freshly downloaded UTXO set (by new nodes) happens by looking at the UTXO-hash in the corresponding block on the chain. Miners cannot include a false UTXO hash in a block because it will be rejected by nodes (This will require nodes and miners to hash the UTXO of the previous block for every new block to mine/verify).
While the UTXO set is 4GB or so, a solution can be made to make it easier for nodes and miners to hash the 4GB UTXO set. Merkle Tree? Keeping most hashes the same every new block, submitting the tree-root UTXO hash of previous block in a new block. Some thoughts were already being exchanged on what the most useful format/data-structure for this would be: https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2017-February/013590.html Note that a Merkle Tree does not need to be a Binary Tree.
In short, the data-structure should be specialized to hash a big file with minor (random?) local changes, I believe such a thing can be made.
 
If this works out, it will eliminate the need for a full-chain initial block download. It also makes it possible to significantly increase block size (since nobody needs to store the full-chain anymore, everyone can be a pruned node).

Picture this: You are a new node joining the network (currently you need to download and process the full blockchain, starting from the genesis block, to verify your UTXO download).
The way to validate your UTXO download, is by looking at the size of the chain going forward. A freshly downloaded UTXO set from 1 block ago will not have much PoW spent on it, thus a new node can be less sure it wasn't modified by an evil node from which he downloaded it. An UTXO set of 500 blocks ago will have much more PoW spent on it. Trust comes from the age of the UTXO state. The older the UTXO state, the harder to modify, since the PoW chain is longer, and the chain contains UTXO state hashes which have to be correct each block.

Another, additional way of adding trust to a downloaded UTXO is to query other nodes if they have the same hash for an UTXO state, corresponding to a block's timestamp, the more nodes confirm, the more trust it would add. (But the primary method is that you don't need others to verify, you can verify yourself).