You've gotten the positive feedback that you should get (and sounds like needed to get 
). Once he gets to the tryout date, yes, it is in many ways "as simple as doing his best, be the young man that he is, and let it all fall in as it may,"
I will, however, offer up the other side of the coin that may or may not apply to your son. All of the great attributes you describe are fantastic. But, at the end of the day, the baseball coach is looking for the players with the best skill set and/or potential skill set. Yes, if there are two comparable players, he will most likely choose the kid with the best attitude and work ethic. But, if, for example, your son is at a large HS with a large pool of good players, attitude and work ethic may not be enough to put him above the cut line if the skill set doesn't measure up. Also, choices have consequences. If a player chooses multiple sports in HS, his immediate development in one specific sport can be affected by the simple math of less time put into practice, sport-specific workouts and games. There are many pieces to the puzzle.
Personally, I am a proponent of multisport participation and it sounds like your son is putting in the extra work to assure continued progression of his baseball skills. In fact, I would guess that he is doing more than most who are baseball only. I also love the idea of swimming for conditioning (although you may hear some negatives along the way with regard to throwing arm health). But, there may come a time when a choice must be made if the proper commitment can't be met in order to reach attainable goals. There may come a time when one coach isn't happy with the inability to fully commit due to participation in the other sport. There may be an instance where being in front of the coach for 10-12 days of tryout is more beneficial than 2 days. There may come a time when your son decides he wants to be fully committed to one sport or can only handle the rigors of one sport while trying to maintain his excellent academic standing. These are all case-by-case scenarios that you and your son will have to monitor for his particular set of circumstances. As others have mentioned, most HS coaches are accustomed to the overlap of HS seasons and providing adequate tryout opportunity to those kids coming in late due to participation in a preceding sport.
You made comment about concern "mainly regarding his primary position". I would suggest that the best attitude he can have going in is that he is 100% ready and willing to fight for making the team, not making the team at his primary position. You say he is working hard on hitting and fielding so I assume his primary position is not pitching. A player who is really valuable to a team is capable and more than willing to play any position. This allows a coach to better put his best nine on the field. If the team has 2 or 3 great SS's or CF's or C's and they are all very capable at other positions, then they can all find a spot in the lineup and the team is better. This may or may not apply to your son at this school right away but if he continues forward in the game, it will be a factor at some point. As example, it is often noted that many colleges recruit largely the best shortstops they can find and put them at various positions in the field.
Your son sounds like a great kid. Bottom line - He has put in the work. He is ramped up and ready for tryouts. Yes, he should simply do his best, be the young man that he is and let the chips fall where they may.
And then, get back to work and face the next challenge. Best of luck to him.