Two reasons you go with Sczur over Jackson
1. Szczur is way more likely to be ok with a limited role getting spare AB/mostly fielding where as Jackson will want to play more to rebuild his value
2. That money you pay Jackson impacts your ability to trade for money later in the season if necessary
There simply isn't a role where I'd say "let's sign Austin Jackson for 6+ million" to be a fourth OF. If I truly was concerned about my lefties splits, I'd play Baez at 3B and KB in LF. Or move Russell up in the order and bat Szczur in the 9 spot.
Again, i don't see a world in which Jackson gets $6 mil+. Whether or not Jackson wants to play more is beside the point. If he signs with the cubs he couldn't get a better offer and therefore had no choice in the matter. Additionally, what you're also missing here is that the cubs don't have a ton of OF depth. Let's say for example that Soler goes down which given his history isn't that surprising. And let's also suggest in this scenario they don't sign anyone else. You then presumably promote Szczur. What happens if someone else gets dinged for 2-4 weeks? You only other option at that point is Baez and Murton who's not played in the majors since 2009. Maybe Baez adjusts to the OF well or maybe he's Hanley Ramirez out there. After him you got a giant maybe in Alcantara who I love but let's be real here he's not proven much. You got Almlora who could be good but chances he's ready by even midseason for the majors are low. You could perhaps throw Zobrist out there but he had leg issues last year and chances are his range is hurt so I'd rather keep him at 2B. And more to the point, they have very limited options in CF particularly.
So with that being said, sure if Jackson get's like $7-8 mil you aren't going to pay a bench player that. But at say $3-4 mil? I can't imagine that is the difference between them making a move later in the season. And frankly, OF depth is a move they made last season at the deadline anyways which clearly suggests they weren't 100% happy with Szczur's performance anyways. Also for what it's worth, Szczur graded out as a -12.9 UZR/150 in all of the OF in 283 innings(-52.3 in CF over 88.2) and -6 in DRS(-5 of those were in CF). So, this isn't a case where he's all defense and no bat either. And in a world where the cubs do make a move for say pitching, there's a good chance that someone like Baez, Soler, Szczur, Alcantara...etc could be included in a deal. Obviously at this point Baez/Soler would be the meat of a deal while the later would be filler but teams often want guys like Ramirez/Grimm in deals who are MLB ready or very close and cheap.
It's kind crazy to me that I'm having to defend bringing in a guy who's played stellar defense in CF and has a career .333 OBP as a bench piece. Like I said I get the idea you don't want to drop a crap ton of money into a bench player but have people seen the calendar? We're literally 6 weeks and 3 days from opening day. You still have a number of very decent OF out there. Fowler, Denorfia, Jackson, Marlon Byrd, Jeff Francoeur, Matt Joyce, Nate McLouth, David Murphy, Alex Rios, Grady Sizemore, Drew Stubbs, Will Venable and Victorino are all still out there. At this point what leverage do those players have to demand a starting job? Looking at fangraphs rankings for CF the 10 worst teams are the Padres(Melvin Upton), Rockies(Charlie Blackmon or Parra), Rangers(DeShields with Lewis Brinson looming), Phillies(Bourjos and Herrera), A's(Billy Burns and Crisp), Indians(Rajai Davis with Michael Brantley on the DL), Tigers(Anthony Gose and Maybin), Braves(Inciarte, Bourn and Mallex Smith looming), and the Nationals(Michael Taylor and Ben Revere). Those are the 10 worst teams in terms of CF and I have a difficult time arguing that any of them should sign Jackson for starter money. A number of those teams should probably be tanking for picks not trying to compete. And those that might compete like the rangers and nats have younger options who are as intriguing.
So like I initially said, I see no way Jackson gets a decent payday. Players in his situation are literally going to be hunting for jobs because as late as it is in the offseason they have no leverage. The full team reports for spring training next week. If you don't have a job by then you're in panic mode because there's a very real possibility you wont next season.