If in 3.5 a monster's CR is intended to be the level at which one PC has a 50% chance of winning the fight, why isn't a monster's CR used as the starting effective character level for a monstrous character of that type?
Because monster CR is based on the utility of its powers given that it's going to live about one encounter, while the benefit of those powers to a PC that exists over many combats is different. This is explained by designer Sean K. Reynolds in detail here. It's also explained in Savage Species (sidebar under "Basics of Monster Characters"):
(...) The factors that go into making a good challenge and a good character are so different that CR is no help in the latter case. A monster is only “on the scene” for a very short while, usually just a few rounds. A player character, on the other hand, is present for almost every scene of the adventure. (...) Never assume that CR and ECL are equivalent, or even related.
You should probably read Savage Species if you intend to start using monstrous PCs as it has all kinds of important guidance of that sort.
You're framing the question wrong.
3.5 is not balanced for 1v1s, and its not balanced for a primarily risk-focused combat encounter. This fundamentally puts you in the wrong frame when evaluating CR vs. LA if you view it through the lens of 1v1s and risk-focused combats.
The game is instead balanced for party-based combat(4 PCs versus things) and resource-management-based assymetric combat encounters*.
Taking this new frame, we are now better able to evaluate the LA vs. CR balance. Let's look at a few examples
Succubus - Is a CR 7 that has many at-will abilities that are either of spell level 3 or 7. Level 3 spells are supposed to be a still-rare resource at APL 7, while level 7 spells are supposed to be an expensive consumable for PCs of this APL. But let's look at what kinds of PCs a Succubus fights. A Succubus is a CR+0 fight against 4 PCs of level 7, so her at-will spell-likes let her maneuver and stay safe, but they don't make her significantly more deadly in a straight fight. If she's with a group, she could reasonably face-off against level 11-13 foes in a group of 4. At this point, she's incredibly reliant on her spell utility to stay relevant, as her stats haven't aged well into this zone, despite it all. A succubus would be broken as an ECL 7 PC, but is merely credible as a CR 7 monster.
Lycanthropes - With a CR increase of +2, vs. an ECL of +4 or more the disparity is dramatic. A werewolf(CR +2 vs. 1 RHD and LA+3) gives DR 10/silver. As an encounter fight, this werewolf will fight PCs of APL 3 or higher. Rogues have 2d6 SA. Fighters can power-attack for +9 or more damage. Warblades can do 4d6+Str or more damage. Even if the melee can't do enough to overpower the DR and don't have silver weapons, the casters can still bypass it entirely with elemental damage. 2 Shocking Grasp are enough to kill it. As a PC, at a base level 5, we have an AMAZING tank. DR 10 to a rare metal means that this PC is immortal against the CR 1 foes he'll fight, and quite tanky against the singular CR 5s, especially because monsters tend towards more attacks rather than more damage. Worse, as a PC with DR, any damage absorbed by his DR is effective healing, saving the party resources in the long run by just not being dealt damage.
*By the DMG, a full 80% of encounters are supposed to of a significantly weaker power-level to the PCs. If the PCs are APL5, a singular CR5(so 4v1) or 4 CR 1s(assymetric power levels).