Design thinking, Lean UX, and Agile are all process frameworks that have been evangelized to the point of religion and adopted in ways that bear little resemblance to the original intent. Here's what each is actually good for and where each breaks down.
Design Thinking
Design thinking (Empathize, Define, Ideate, Prototype, Test) is valuable as a mindset and a teaching tool. The five-stage model helps teams who are too solution-focused slow down and spend time understanding the problem. Where it fails: when treated as a linear process, when the "empathize" phase is too shallow to produce genuine insight, and when "prototyping" is interpreted as the deliverable rather than the learning tool.
Lean UX
Lean UX applies lean startup thinking to design: form a hypothesis, design the minimum experiment to test it, measure, learn, repeat. It's valuable for teams that need to move fast without building the wrong thing. Where it fails: when "lean" is used as an excuse to skip research entirely, or when the metrics being measured don't actually reflect user value.
Agile
Agile is a software delivery framework that prioritizes working software over comprehensive documentation and responding to change over following a plan. It works well when design is integrated into the sprint process, not treated as a separate phase. Where it fails: when sprints are so short there's no time for research, and when "agile" becomes an excuse for never looking more than two weeks ahead.
What You Actually Need
You don't need to pick one. The teams I've seen work best mix all three: design thinking for framing new problems, Lean UX for running fast experiments, Agile for delivery. The skill is knowing when to apply each.