Having stakeholders "endorse" blocks has, according to you, the benefits of increasing the number of full nodes and making a 51% attack more expensive. It seems to me it would have the opposite effects and other negative side effects. Any stakeholder that has "won" could just be running an SPV client and be informed by a full node that they have won, then cooperate to collect the reward. You are mistaking proof of stake as a proof you are running a full node. At the same time, the network becomes cheaper to attack in proportion to the amount of the block reward that is paid to "endorsers". Another side effect is that miners would have a bigger economy of scale. The more stake a miner has, the more they can "endorse" their own blocks and not others blocks. I recommend reading this: https://download.wpsoftware.net/bitcoin/pos.pdf

-Andrew Lapp