What advantage is there for a baseball player to be selected with a higher draft slot if he signs for "below slot" money?

Draft slots are essentially irrelevant in baseball except for the limited purpose of setting an opening salary (which is by convention). After the initial contract, it will make absolutely no difference. There is no advantage in free agency due to draft position. Any future contracts after the first will be based on his production, not the perception of him when he was drafted.

As a sport baseball has one of the worst drafts. Good players frequently are taken in later rounds and early choices frequently don't pan out. For example, previous Pirates first overall choice Jeff King was an average starter, with a best of 30 home runs in a year at a time when McGwire and Bonds were chasing the Ruth/Maris records (60/61). The best of the four previous Pirates' first overall picks was Gerrit Cole, followed by King, Kris Benson (below average but an eleven year career), and Bryan Bullington (ten decisions in five years; left the MLB for Japan). None of the four finished their career with the Pirates. Cole is the only serious Hall of Fame contender.

None of the four 2020 Hall of Fame inductees were first overall choices, although Derek Jeter and Ted Simmons were first round picks. Larry Walker was an undrafted free agent. Marvin Miller was inducted for his front office contributions, not playing.

Between college and the minor leagues, draft picks are risky. The best players may be in high school, while lesser players are finishing college. And regardless, most players spend at least a year or two in the minors before they make it to the major league. Basketball and football players start in the NBA (although that's changing) and NFL respectively, and drafting out of high school is much rarer.

It is quite possible that the Pirates would be better off signing all their late round choices than a first overall.

Assuming your source is correct, Davis presumably expressed a willingness to sign a "below-slot" contract because he still expects to make more money that way than if he had waited until his natural position in the draft. One draft prediction put him fourth. The slot values:

1.  $8,415,300
4.  $6,664,000
-   $1,751,300

However, for all we know, he might have fallen farther. His agent would know more but might not be sharing. Note that that difference would be almost enough to sign the Pirates' second round pick at slot value ($1,999,300). Of course, he might not have signed at that level. We don't know what they have discussed. Perhaps he will get paid the $7,789,900 for being drafted second overall or something else. That would be enough savings for their fourth round choice.

CBS Sports also had him as the fourth best player.

It's also worth noting that the other top prospects that the Pirates were considering might not have signed at all. As high school players, Marcelo Mayer and Jordan Lawlar could refuse to sign, go to college, and then go through the draft again.

It's also conceivable that he preferred the Pirates to other teams who expressed interest in him. He's from New York, which is adjacent to Pennsylvania (where Pittsburgh is). He had previous discussions with the Pirates when he graduated from high school. They did not draft him then because they weren't willing to pay him what he wanted to forgo college.

It also may be that other teams that were interested in him also wanted to pay below-slot. In general, the top drafting teams need to fill a lot of positions. If he were insisting on slot value, he might have fallen below fourth.