This kind of attack works because the spammer sends spam at a much faster rate than the Bitcoin network with its fixed 1 MB blocksize can possibly mine. There is consequently little or no cost to the spammer other than the spammer's own bandwidth. There is a significant cost to the network; however in driving up the bandwidth costs of those of us who run full nodes, with the likely result of driving down the number of full Bitcoin nodes.

A ~60% effective blocksize increase via say SegWit alone will not address this kind of spam attack.