Will taking programs at the Army War College or Command & General Staff College increase my promotion chances and to head army formations?

The school requirement for Reserve Officers is the same as for Active Duty. There are specific formal schools required to reach the next rank. As an example of the progression, after you commission, you will attend Officers Basic Leadership Course. Between that and the next formal requirement is Branch detailing, but then as an O3/ Captain, you will be required to attend the Captains Career Course. I've been out a long time, but I believe the remaining sequence is Intermediate Staff College, Advanced Military Studies and then Senior Service School (read War College). In addition, you will be expected to have picked up a Masters Degree along the way to remain competitive. So the short answer is that these courses are mandatory for you to advance to the level you are talking about.
This is in addition to broadening assignments, various staff rotations and various service schools that your job may require, but aren't technically required for promotion.
So if you want to commission, expect to do a lot of schooling for the rest of your career.

All are required should you be fortunate enough to hit varying levels of leadership and command. Should you refuse any one program you would essentially vault your career. The knowledge gleaned through all leadership programs have proven the test of time. Officer or enlisted, none of us would get to the top levels of the military without the benefit of leadership training and development.

You just don't randomly take classes at the Army Command and Staff College. You are selected for Resident Imtermediate Level Education ILE or nominated for the correspondence or Common Core versions. Concurrent with the resident course or successful completion of the ILE requirement, you may apply for the School of Advanced Military Studies (SAMS). These courses are the mainstays of the CGSC. They also conduct the Pre Command Courses.

So when I was a young officer, it was a selection process. 50% of your YG was selected for CGSC, the other half non select and at risk for promotion to LTC. Non selects had to do a distance learning version. Then the right around my promotion to Major, everyone had to go and no selection board. This proved untenable due to class seats. I understand it is now back to a selection process and a nonresident version and a hybrid version known as common core. I competed for and was selected for a sister service course instead.

The way it worked is if you were non-select, you were 50% chance of not making LTC. There were a small number of non-resident CGSC officers that were promoted. Like 10-15%. So yes to be promoted to LTC, you need resident ILE/CGSC.

Likewise, you simply don't take a class or two at the Army War College. Selection for AWC and it's defense and sister service equivalents (Senior Staff College-SSC) is highly competitive. For command tracked officers, you don't get selected for SSC, you will not get selected for BDE Command. You compete for SSC generally as a post Battalion Command O5. I don't know how a 73A Social Worker competes in the process, but it won't happen realistically until you are a KD complete LTC.

What should you worry about? BOLC, any Army schools they send you to, and the Captains Career Course. You should check DA PAM 600-3 for the career progression and the Milsuite career map for 73A. It will tell you what jobs, when, and what schools. Circa 2016 they separated the branch specific from DA PAM 600-3 and put it on AKO in a Milsuite page, so I can no longer see it. It's not listed here but others are: https://medicalservicecorps.amedd.army.mil/leader_dev/aoc_career_paths.html