The JIS, ISO and ASME standards for cross head screw drivers are fully compatible.
The myth that JIS needs special screwdrivers is caused by people using either using the wrong size (easy at small sizes), using a worn-out or poorly-made screwdriver, or using Pozidriv (which is genuinely very different).
Anecdotally, about 5 years ago I bought a set of off-brand "premium" screwdrivers, where it turned out the Phillips head ones (yes there are two L's in there) were so far outside the standard I had to just throw them in the trash.
You got the wrong standard. JIS B 1012 type H is Philips equivalent, but what people refer to as the JIS screw is JIS B 1012 type S. JIS B 1012 Type S ("JIS") and ISO 8764 (Philips) are entirely different screw and scredriver profiles. See https://www.peterverdone.com/jis-cross-head-screws-and-drive....
The Philips screwdriver has lobes with a large multi-stage taper that ejects the screwdriver from the screw upon torque. The JIS screwdriver lobes are in contrast almost perfect 90 degree angle with a tiny filet in the center - not dissimilar to the Frearson screw.
A Philips screwdriver will be unable to insert all the way into a JIS screw due to this taper, exactly similar to how the extra lobes on a Pozidriv screwdriver collides a Philips screw. Attempting to apply torque with this partial engagement puts you at very high risk of stripping the screw.
A JIS screwdriver however does work perfectly fine in Philips screws, often even better than the Philips screwdriver as you did away with one half of the "screwdriver ejection feature".
There's a reason that iFixit goes out of their way to make both JIS and Philips bits, and warn you in their guides. They aren't really in the business of trying to cram more bits down your throat for fun.