There is nothing in the Matthew context to suggest Herod should be the referent. In fact, there is evidence in the context to support the fact that Herod did not wish to kill JB. Mark backs this up on several points.

Note, fyi, that Herod feared JB, himself, ‘knowing him a man just and holy’, but it was Herodias, the blood lusting wife who feared the crowd. Herod didn’t fear the crowd because he was not planning on killing JB to begin with. It seems he liked the guy on a certain level.

Hope you enjoyed my opinion.
Peace.

This makes more sense if he likes John – though admittedly Matthew hasn’t seen it that way, since he more or less copies the verse into his own account.

I suppose you can say Herod was “grieved” that he had to upset the people…

If it means that Herod arrested John at the instigation of Herodias, then the passage as a whole does consistently present Herod as John’s protector. It would then be difficult to harmonise Mark with Matthew, where Herod wants John dead, because there would be no evidence in Mark of Herod being hostile to John (but plenty of evidence to the contrary).

If it means that Herod arrested John because of Herodias, i.e. that Herodias was the reason for but not the instigator of the arrest, then the arrest can be read as evidence of Herod being hostile to John. The idea that in both accounts Herod wants John dead but doesn’t kill him for political reasons would then look more plausible.