Sparta and Rome suffered from the same problem, actually. They approached it differently. Rome survived. Sparta didn't.
The problem was that citizenship or social class was based on inheritance and it was essentially a closed system. At a time when infant mortality was so high and being in the military was risky, this meant that the ruling class diminished over time.
The author of this piece touches on how most soldiers had to bring their own gear. That's fair because they're talking about why people fight not how but it's important. So in Rome, you were cavalry if you could bring your own horse. Only the wealthier people could do this. This was a later reform. The very early Roman cavalry were a closed class. We don't know a lot about this other than legends because records were lost, most notably when Rome was sacked by the Gauls in 390 BCE.
Anyway, so "equites" were a higher social class but were a mix. Some of them were patricians (probably descended from the original heavy cavalry that protected the pre-Republic King, allegedly). Also, some patricians weren't equites.
In the early Republic there were 40-45 Patrician families. They held all the political power and offices. By Caesar's time it had dwindled to ~12, so much so that some plebeian families got elevated.
Sparta was what we'd now call an ethno-state requiring a full bloodline for citizenship. AFAIK Sparta never evolved away from that and all such closed societies die, just like the original Roman patrician families.
Rome took a different approach that bears some similarity to "whiteness" in the modern world. Race is an invented social construct (and, in the case of whiteness, was invented to justify slavery). But what has made whiteness resilient is that definition of who is white keeps evolving as necessary to maintain the in-groups.
For example, Ben Franklin didn't consider Germans "white" (famously describing them as "swarthy" [1]). Through different waves, different European ethnic groups became "white". The Irish didn't become "white" in the US until 100-150 years ago. Arabs became "white" in 1915 [2].
I just want to stress how made up this all is. Anyway, back to Rome.
Rome was famous for taking over an ethnic group (usually quite violently) but then making them Roman. Many people on the periphery of the empire aspired to become Roman. We have a term for it: Romanization
[1]: https://medium.com/@cailiansavage1/why-benjamin-franklin-did... (or Latinization). There's historical record of this everywhere from Eastern Europe to Northern Africa ro Britannia. Britannia was a funny one because there are Latin inscriptions describing a very Roman life from people who unsuccessfully rebelled against Rome in the early occupation a century earlier.
So I guess I'm saying is that yes, Sparta atrophied as all purity-based ethnicities always do whereas Rome survived much longer with an expanding concept of "Roman-ness", which isn't too dissimilar from what we recognize as "whiteness" today.
[2]: https://teachinglegalhistory.unl.edu/s/oer/item/1999