Background: Alzheimer’s disease (AD) is a progressive and irreversible neurodegenerative condition that increasingly impairs cognitive functions and daily activities. Given the incurable nature of AD and its profound impact on the elderly, early diagnosis (at the mild cognitive impairment (MCI) stage) and intervention are crucial, focusing on delaying disease progression and improving patients’ quality of life. Methods: This work aimed to develop an automatic sMRI-based method to detect AD in three different stages, namely healthy controls (CN), mild cognitive impairment (MCI), and AD itself. For such a purpose, brain sMRI images from the ADNI database were pre-processed, and a set of 22 texture statistical features from the sMRI gray-level co-occurrence matrix (GLCM) were extracted from various slices within different anatomical planes. Different combinations of features and planes were used to feed classical machine learning (cML) algorithms to analyze their discrimination power between the groups. Results: The cML algorithms achieved the following classification accuracy: 85.2% for AD vs. CN, 98.5% for AD vs. MCI, 95.1% for CN vs. MCI, and 87.1% for all vs. all. Conclusions: For the pair AD vs. MCI, the proposed model outperformed state-of-the-art imaging source studies by 0.1% and non-imaging source studies by 4.6%. These results are particularly significant in the field of AD classification, opening the door to more efficient early diagnosis in real-world settings since MCI is considered a precursor to AD.