A sweeping independent review of the Department of Veterans Affairs health-care system made public Friday shows the multibillion-dollar agency has significant flaws, including a bloated bureaucracy, problems with leadership and a potentially unsustainable capital budget.

More than a dozen assessments—from analysts including Mitre Corp., Rand Corp. and McKinsey & Co.—show that the Veterans Heath Administration, the health-care arm of the department known as VHA, is still plagued by long-standing issues, including unsustainable...