This is a frustrating but fairly common issue with laptops of a few years old, and the intermittent blank screen on startup is a useful clue. Here are the most likely culprits and what to try:
Battery and power delivery
Even though you have already tried the drain-and-hold trick, a failing battery can cause exactly this kind of unpredictable behaviour - working fine sometimes, refusing to start other times. Try running the laptop with the battery completely removed and only the AC adapter plugged in for a few days. If it starts reliably every time, the battery is the likely cause and replacing it is usually inexpensive for a Lenovo.
Overheating and dust
If the laptop is a few years old and has never been cleaned internally, compacted dust in the vents can cause the machine to struggle on startup, especially if it thermal-throttles or hesitates before the fans can spin up properly. A can of compressed air directed into the vents (with the laptop off) can help, or if you are comfortable opening it up, a soft brush around the heatsink and fan makes a real difference.
RAM seating
A loose or slightly oxidised RAM module can cause blank-screen startup failures that are maddeningly intermittent. It is worth opening the bottom panel, removing the RAM stick or sticks, cleaning the gold contacts gently with a pencil eraser, and reseating them firmly. This costs nothing and fixes the problem more often than you would expect.
If none of those resolve it, the next suspects would be the CMOS battery (a small coin cell on the motherboard, cheap to replace) or a failing display cable connection, though the latter usually shows differently.
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