Problem with the L298 datasheet

Those old bipolar processes could only be used to make crummy lateral PNP transistors with quite low gain so not very good for high current.

Using all NPN, especially those NPN, transistors is bad (because of the extra Vbe drop), but not as bad as it would be with monolithic PNPs.

Popular? Probably for legacy applications and hobbyists, but I doubt any serious application has designed them in within the last decade or more.

In your H-bridge schematic, the same signal is being used to drive the bases for the upper and lower transistors. Because the top ones are PNP, and the bottom are NPN, the tops will conduct when the signal is low, and the bottoms will conduct when the signal is high.

The L298 schematic shows that inputs for the lower pair are inverted. That's the little circle in the inputs to the AND gates. So even though the tops and bottoms are all NPN, the tops and bottoms will still conduct at opposite times.

The schematic is correct.