Yet, the elephant in the room is the nothing at stake issue at this point. Or seems to be anyway.
Nothing at stake is a strawman. Include a recent block hash in every transaction and allow migrating transactions cross-chain ("invalid" block headers in tx don't count for stake-vote). There's your motive for picking a chain.
You haven't eliminated the fact that there is absolutely no reason not to sign all possible chains in event of a fork. It is just a small step in the right direction, albeit at the expense of larger txs. Bitcoin in it's short history had had ~40M txs. Adding 32 bytes to each transaction would have bloated the blockchain by another 1.2 GB. If that solved the problem it probably would be worth it but it doesn't so one has to weigh if that increased security is worth that increased cost (in terms of tx size & complexity). I think it probably is but it isn't a magic bullet and it isn't free.
Nothing at stake covers a number of different scenarios that all have at their core are related to the fact that there is no (significant) cost for failure*. One can perform a reorg with no cost by simply acquiring private keys which at one time had (past tense) a majority of the active stake. Accounts which have no value have worth to an attacker for what they once had. Nothing at stake also applies to the concept of selfish mining. There is no reason to not continue your chain when you lose a race against a superior block. You can work on the majority chain and simultaneously attempt to extend your minority chain. There is a small chance you will out mine/mint/forge the rest of the network. In PoW the cost doesn't warrant the small chance and thus it is more profitable to switch when you lose but if there is no cost why wouldn't you keep trying, you have nothing to lose and everything to gain. In NXT when you forge a block which is the best at current height but doesn't enable you to forge the next block (due to incompatible block signature) there is no reason to not use computing power to try and find an alternative which enables you to forge the next one as well. The chance may be low but you have nothing to lose and everything to gain by trying. With PoW there is a real cost associated with extending an inferior chain (one that has a less than 50% chance of remaining the longest chain) and that cost drives the network to reach a consensus quicker. Inferior chains are abandoned and the network aligns around the best chain quickly. How often have you seen reorgs more than 3 blocks deep in the blockchain? This alignment doesn't happen out of some need to do the "right thing" but because it is prohibitively expensive to do anything else. With no cost, forks will last longer, and reorgs will be deeper. In any blockchain system, attackers with a minority of the stake can (infrequently) generate short lived chains that are better than the majority. With PoW those attempts have a real cost which counteracts the gain of the double spend and unless the attacker has a high chance and profitable spend it is very likely a net loser to even try. When the cost of an attempt is nothing there is nothing to lose by trying and thus why not double spend anything and everything. Most of the time the attacker will fail but those failures cost nothing so any successful double spends is profit even if rare.
So saying "put a hash of a recent block in the tx and nothing at stake is a strawman" misses the larger context. If there is no cost to attack the network there is no reason to NOT attack the network.
* Yes some will like to split words and point out there is always some cost in anything so saying there is "nothing" at risk is wrong. Well you can split hairs all you want but if a legit node has a cost of X and an attack node has a cost of 10X it is essentially nothing. X has to be pretty low such that non mining nodes have the resources to run a full node so 10x is still very low. PoW doesn't require 10x the resources to solve a block than to run a non-mining node, or even 100x, or a billion times but (at current difficulty) 193,690,812,773,950,000,000,000,000,000x.