tl;dr: you could make data publication much more expensive by requiring transactions to prove that data in them isn't arbitrary. But even that will not totally stop data publication. Also, not all “spam” requires data to be published.

You would have to try to forbid the spam via consensus rules, i.e. soft fork it out.

This would very quickly boil down to us choosing between having a flexible scripting system or not having spam transactions. Even if we restricted ourselves to whitelisting only specific single-sig payment schemes and forbidding all other transaction types, it would be possible to embed data in other transaction fields like signatures, public keys, or pubkey hashes. The trade-off seems rather unattractive to me.