I'm not proposing that individual people would have to submit PoW, but instead that individual transactions would. Thus, a single transaction with many hundreds of outputs or inputs would require a relatively large PoW, but a transaction where the entire balance of one address is transferred to another address would not. (This has been my working definition of spam: transactions of the former type rather than the latter.)
I don't think spam transactions have been large transactions, rather they are small ones and there are lots of them. So this wouldn't help. Perhaps if you inverted it, smaller transactions have a higher PoW. But then spammers would just make large transactions. And if you penalized large transactions, they'll make smaller transactions.
Furthermore, as I said earlier, getting an ASIC or a GPU or two would basically make this completely useless unless you can somehow adjust the difficulty to the volume of transactions a person is creating. If I were a spammer, I could get an ASIC and just churn out spam anyways because the ASIC is going to blow through whatever difficulty you set unless that difficulty is specifically being adjusted for my spam.
Now you might say that getting an ASIC or GPU is costly and thus prohibitive, but it has been theorized that many of these spam attacks are coming from miners who want to increase the fee rate and thus increase their mining profits. So if a miner is mounting a spam attack, they can take a single ASIC machine that is being used for mining and repurpose it to get around the PoW limitation. They would still be able to churn out spam transactions as quickly as before.
Lastly, as ranochigo said, your proposal would have a detrimental effect on the UTXO set. You would be encouraging the creation of UTXOs when what we want to do is encourage the consumption of UTXOs. By making large transactions costly, people will make smaller transactions and likely with change (so likely 1 input and 2 output transactions) which will result in a net increase of UTXOs. This is not very good for Bitcoin, and it has the effect of grinding down one's UTXOs into dust thus making it hard to spend any Bitcoin later.