I feel like the memorization just comes naturally?
Indeed. Spending time understanding compositional architecture, how sections of the piece are organized and assembled to make the whole, repetitive patterns in rhythm, harmony and melodic line. All of this aids in memorization. Do the appropriate steps to understand the composition, and most of the memorization will be done under the auspices of that process. If you know where you are going you will get to the destination much more efficiently as opposed to wandering around haphazardly under the false pretense of serendipity in hopes that you will get there eventually.
As for tempo, that can be streamlined to only required two settings: practice tempo and performance tempo. There is no explicit need for a gradual speed up. You can do a compositional analysis at the keyboard at performance tempo. In essence you will be training yourself to think at performance tempo from the very start. What remains is to learn the physical aspects of playing the piece, and bring that in line to your cognitive understanding of the music.
Gain enough experience efficient learning techniques, and you will develop the ability to sight read at tempo. Orchestral and choral musicians tend to develop this ability much earlier than pianists. Unfortunately, for pianists, ensemble playing is not emphasized enough in the earlier years of study.
Be methodical and regimented, you can do this.
Topic: Learning Chopin's "Valse Brillante" in 2 Months