I think the Chiefs are trying to make a run during the 4-year rookie contract window they have left on Mahomes. While they don't expect to suck this year, they think they are at least a year away from competing for a SB. While a lot could happen for other teams, that window may also coincide with the decline of Brady and the Steelers blowing up.
As short as they are on draft picks, I don't think they'll move up despite Veach talking about how aggressive he likes to be. I think they come away with 3 players in the first 3 rounds, but I wouldn't be surprised to see them trade a pick for a current player. In the 3rd, the Chiefs could still look to get potential starters/rotational players at G, C, ILB, RB, slot CB, or SS. Not premium positions but spots where the Chiefs have a need.