What are the main reasons why a 3-Star Flag Officer doesn’t end up becoming a 4-Star?

Short answer: Yes.

Long answer: The number of positions is limited to both law and Congressional caps (10 USC 525/526*). By law, the services have a maximum number (i.e,. for the Army, they can't have more than 46 total O-9s/O-10s of which no more than 8 can be O-10s) which totals up for DoD to be no more than 148 O-9s/O-10s of which no more than 27 can be O-10s.

However, there are a number of situations where the general and flag officer (GFO) doesn't count against the max (for example, specifically the Chief of NGB doesn't count against the statutory cap on O-10s as well as up to 19 4-Star Joint Duty Assignment positions). That's why although the Army is limited to 8 O-10s, they actually have 15 on active duty (as of 30 SEP 23).

Of note, GFOs are not technically 'promoted' to a 3-Star or 4-Star rank (they are ceremonially promoted of course), because the highest permanent rank that an officer is boarded for is 2-star. The 3&4-star ranks are by appointment to that position and that rank holds as long as they are in that position and reverts back unless they are appointed (and confirmed) to another position.

Another note - because the 3-Star and 4-star positions are by appointment, GFOs can "skip rank". Although rare, it does happen (last I know of is Brig Gen. Conley that is being put up as the O-9 commander of AFSOC).

When an O-9/O-10 retires, they CAN hold that rank, but unless approved by Congress (it almost always is), they revert back to the highest permanent rank they held (10 USC 1370*)
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* 10 U.S. Code § 525 - https://www.law.cornell.edu/uscode/text/10/525
* 10 U.S. Code § 526 - https://www.law.cornell.edu/uscode/text/10/526
* 10 U.S. Code § 1370 - https://www.law.cornell.edu/uscode/text/10/1370

Number of open slots' and sometimes- officer politics.

There are always very limited numbers of slots for General/Admiral (O-10). I worked around general/flag officers for several years. Some would talk a little about the general officer promotion system, but most wouldn't. In short, it's unlike the enlisted or officer promotion systems. Yes, general officers are evaluated, but many other factors come into who gets the top slots. IMO personal relationships make a big difference. Very senior officers may tend to select officers they know and trust for O-9 and O-10 positions they control. Once in the job, the promotion to the appropriate grade is pretty well assured. Additionally, many of these positions are controlled by very senior political appointees in the Service or DoD. Presidential appointments to positions at the most senior levels, such as Service Chiefs, are very political. Reputation and "who you know" are definitely factors. I suppose in today's world, Diversity, Inclusiveness, and Equity play a part also. Glad I missed that.