Background/Objectives: Alzheimer’s disease (AD) is a progressive neurodegenerative disorder for which MRI-based AI systems are increasingly used for diagnosis and prognosis. However, many published approaches remain misaligned with the requirements of trustworthy clinical use. Predicted risks are often poorly calibrated, explanations are frequently limited or non-actionable, guideline-aligned reporting is uncommon, and longitudinal prediction is inconsistently evaluated. In this paper, we conduct a PRISMA-guided structured review with scoping-style breadth of MRI-centric AI methods for AD diagnosis. This design supports a theme-based synthesis across heterogeneous study designs and is intended to summarize the current evidence base and derive practical design requirements for next-generation, clinically oriented pipelines that integrate calibrated staging, explainable outputs, and longitudinal risk modeling. Methods: Searches were conducted across Scopus, PubMed/PMC, and arXiv/bioRxiv (2014–2026; English; human AD/MCI imaging) and were supplemented by backward and forward snowballing. These searches yielded 2460 records. After deduplication, screening, and full-text eligibility assessment, 90 papers were included in the final synthesis. The included literature was organized into thematic streams spanning counterfactual reasoning and explainable AI (XAI), vision–language approaches for report and caption generation, longitudinal and survival-style modeling, and multimodal fusion and transformer-based methods combining MRI with clinical variables and other biomarkers. Vision–language methods were considered together with retrieval-augmented paradigms. Results: Key findings are that the field has shifted toward transformer architectures and multimodal fusion and shows increased interest in richer explanation mechanisms. Nevertheless, calibration metrics and robustness checks are inconsistently reported, external site-held-out validation and subgroup analyses remain relatively uncommon, and guideline-aligned structured reporting with explicit numeric provenance is rare. Vision–language and retrieval-augmented reporting methods are far more mature in general radiology than in AD MRI, highlighting a translational opportunity. Conclusions: Based on these findings, we recommend standardized reporting of classification calibration and longitudinal risk calibration, stronger site-held-out validation with subgroup robustness evaluation, clinically meaningful counterfactuals, and guideline-aligned reporting with reproducible numeric provenance embedded within reproducible pipelines.