1- are you trying to load FMJ or Hollow Point ammo? It was designed for the military's FMJ rounds- as hollow point wasn't 'cricket' according to Hague Accords/Geneva Convention. Hollow Point ammo may stick on the feed ramp/chamber ramp transition, as the ogive is too wide for the design and it thus 'hangs up'
2- If you remove the barrel and 'drop' a complete FMJ cartridge into the chamber, it should drop in all the way. If it sticks, I'd try 2 other brands of FMJ .380 ammo. IF they stick, I'd try to clean/polish the chamber to see if there are any deposits causing friction. Do NOT hone/widen the chamber- clean and polish lightly to avoid changing chamber dimensions.
Sometimes gunk/powder/lead/rust has speckled the chamber, and this adds resistance to the round.
IF the rounds drop into the barrel chamber with no issues, reassemble, lubricate all rotating/sliding surfaces, and try the slingshot load.
If it still will not chamber a round- have it inspected. It is NOT working as designed. You should never need to hit the slide to load a round.
3- verify that your magazine is for the cz 24. If it is for a cz 27, the feed lips may be just a touch narrower, causing difficulty in releasing the cartridge when the slide tries to load it into the chamber.
Also, the correct pistol magazine may have had the feedlips bent at some time. The gunboards forum has a section for Czech firearms and there are a lot of people with experience with the older stuff over there.
4.- have the extractor examined to verify that it allows a round to slide up the breech face under it properly. If the extractor is seized [gunk/poor maintenance/etc] or the tip is buggered up, the round may not slide under it properly, which causes the slide to hang up.
If extractor is 'stuck', disassemble, clean carefully, source a new spring, and reassemble- trying it again.
5. Do NOT put a round in the chamber and then drop the slide. This is a good way to break the extractor, as it is forced to bend out over it. Put a round in the magazine, put the mag into the pistol, and then pull slide back [and let 'er fly] to load it. This pushes the round up under the lip of the extractor.
6. Do NOT dry-fire this pistol without a snap cap. I'd use a snap cap AND an o-ring over the Firing Pin. Older pistol designs/metallurgy/heat treating, when combined with dry firing without the above precautions, can lead to firing pin tips breaking off.
I have NEVER had this problem with my CZ 24. However, I also haven't tried it with hollow points or flat nosed .380 rounds. And I detail stripped, cleaned, lubed, and reassembled the pistol before I ever shot it. [I also did sear/hammer/trigger work to smoothen/lighten it up. Now it has a crisp break around 4lbs.]
As you found, the CZ24 was issued in 9x17mm [aka-.380ACP, aka 9mm Browning Short, etc]. The designed originally intended it for 9x19mm/9mm Luger, but Hungary said 9x17mm. I have never heard of anyone with one of the 9x19 test barrels- but there may be one or three out there on the planet.
The CZ 24 frame was used with a modified top end to produce the CZ27 pistol in .32acp.
The slide shape is the easiest give-away as to which caliber it is: rounded sides and it is .380ACP, while with flat sides it is .32ACP.
The .380 based 24 used a rotating barrel design to delay the recoil impulse. The slide had to be rounded to allow the locking lugs and chamber to rotate.
The .32acp based 27 didn't need that- as it was a blowback design.
Enjoy- they are fun!
The CZ 24 is my daughters' favorite .380 that I have, and probably her favorite semi-auto handgun overall.
I really like it- as the recoil is so much better than the blowback designed .380 handguns out there.