Open-domain video platforms offer rich, personalized content that could support health, caregiving, and educational applications, but their engagement-optimized recommendation algorithms can expose vulnerable users to inappropriate or harmful material. These risks are especially acute in child-directed and care settings (e.g., dementia care), where content must satisfy individualized safety constraints before being shown. We introduce SafeScreen, a safety-first video screening framework that retrieves and presents personalized video while enforcing individualized safety constraints. Rather than ranking videos by relevance or popularity, SafeScreen treats safety as a prerequisite and performs sequential approval or rejection of candidate videos through an automated pipeline. SafeScreen integrates three key components: (i) profile-driven extraction of individualized safety criteria, (ii) evidence-grounded assessments via adaptive question generation and multimodal VideoRAG analysis, and (iii) LLM-based decision-making that verifies safety, appropriateness, and relevance before content exposure. This design enables explainable, real-time screening of uncurated video repositories without relying on precomputed safety labels. We evaluate SafeScreen in a dementia-care reminiscence case study using 30 synthetic patient profiles and 90 test queries. Results demonstrate that SafeScreen prioritizes safety over engagement, diverging from YouTube's engagement-optimized rankings in 80-93% of cases, while maintaining high levels of safety coverage, sensibleness, and groundedness, as validated by both LLM-based evaluation and domain experts.