French press needs a coarse, even grind — coarse so the mesh filter holds it back, even so the cup does not turn silty. It is the least demanding brew method on fineness, and the most obvious about inconsistency.
The metal mesh only stops particles above a certain size. Everything smaller passes through and keeps extracting in your cup, which is where the sludge at the bottom and the creeping bitterness come from.
A grinder that produces a coarse setting with few fines is doing the one thing French press asks. A blade grinder cannot: it makes dust and boulders at the same time, and the dust always ends up in the cup.
Range that actually reaches coarse. Some espresso-focused grinders top out finer than French press wants. Check the coarse end, not the fine one.
Few fines at the coarse setting. This is where burr quality shows more than at any other setting.
Capacity, if you brew a full press for more than one person.
You do not need stepless adjustment, single dosing, or a large flat burr set. French press is the method where the least expensive competent grinder does the job.
If French press is all you brew, buy for the coarse end and spend the difference on coffee.
If you also make espresso, the requirements pull in opposite directions — fine and repeatable versus coarse and clean. A grinder with a genuinely wide range handles both, but no single grinder is optimal at both extremes.
Be honest about which method you brew most, and buy for that one.