The definition of "conservative" by historians seems to generally be the universal training of Spartan citizens to consider and fight for the interests of the state over consideration for personal wants. Their government was set up originally to retain these ideals. What are specific moments during the Persian & Peloponnesian Wars as well as leading up to, during, and immediately after the Battle of Leuctra in 371 BC that displayed these foundations were flawed, leading to negative outcomes for Spartan hegemony? I'm not looking for an answer to all of these at once since that is daunting, I'm fine with reading whatever you want to focus on.
Example questions: 1. Do we have evidence a king of Sparta, who so far as I've read is the sole communicator with the Oracle of Delphi, ever lied about a prophecy in order to push Sparta to go on a military campaign? 2. Moments where the passing of laws by the Gerousia and/or Assembly (or vetoing the Assembly by the Gerousia) lead to long term negatives?
The question as you've asked it here is indeed daunting, and probably unanswerable. There is a tremendous amount we don't know about the workings of the Spartan system. All of our information comes from outsiders, some of whom were writing centuries after the fact, and many of whom had a particular point to make about Spartan character or the Spartan system. One key point with regard to your question is that we do not possess evidence of even a single law passed by the Spartan Assembly in the Classical period, except insofar as they can be reconstructed from later traditions, which make them almost impossible to date accurately. Another key point is that the characterisation of Spartan society as "conservative" or backward is controversial; current scholarship suggests that far from retaining immutable institutions and customs, Sparta was constantly adapting and changing, only retaining an outward fiction that its laws had been perfect from the start.
It would be helpful if you could confirm my impression that what you're actually looking for is essentially three things:
Evidence of individual decisions made by kings, commanders or institutions that (inadvertently) hastened the decline of Spartan power.
Examples of systemic problems caused by the laws of Lykourgos that made Spartan power unsustainable.
Possible connections between the two, that is, ways in which systemic problems caused individual actors to favour bad policies.
If this is roughly what you mean, I can link you to some old answers, but also feel free to have a look through my flair profile section on Sparta.
I'm not sure if this is really relevant to your question since its mostly about the military structure of the spartans rather than their political structure (though I think there's a bit of an overlap here, especially with there reputations) and its a really great post if you've got any interest in the spartans anyhow
its written by /u/iphikrates who also just was featured on the latest episode of the askhistorians podcast and might be able to shed some more light on your question though his flair is on greek warfare so maybe not hah