This is a very accurate point, which can be fixed , if personal greed does not prevent it.
If a % of each BTC reward or a % of the BTC transactions were shared with anyone operating a Full Node, that would be incentive enough to keep active full nodes.
Something similar to Dash MasterNode's system would work.
But Core has proven , they are looking out for their personal interests and not the general interests of the BTC community.
FYI:
Since Proof of Stake was mention trying to insult me, here is how using Theymos idea of adding proof of stake could help the node problem.
If PoS was added to BTC, any full node could generate a PoS block and Receive the Transaction Fees for including them in the block.
At the current environment , that would be ~$1000 of fees for every block staked.
Now that is Incentive to run a Full Node!
The good thing with decentralized consensus, that we still have, but not for long if this continues, is that we can outvote the miners, if all nodes put their heads together.
Now the challange is that it's hard to introduce change in a decentralized enviroment, and that is the whole point of it, to stop malicious entities from hijacking Bitcoin.
The problem is that this would require BTC to be perfect before launch, so that no additional uppgrades would be needed.
Of course this was the fault of satoshi, of starting a flawed network in 2009, he should have waited some time and find some programmers to think about it.
With 1 more year of engineering and bugfixing, bitcoin could have come out flawless.
But now we have to live with a flawed version of it, that has 16 billion market cap, and it's immutable. How are we going to fix the flaw now?
How are we going to get consensus in a decentralized system, where the point is to not have consensus, to prevent malicious hijacking.