how to remove old painting from laundry floor, if I can't grind it before repainting
Welcome to the Bunnings Workshop community @Margit. It's fabulous to have you join us, and many thanks for your question about re-painting your concrete floor.
It sounds like you've discovered that new concrete paint needs to be painted over either bare concrete or sound previous paint. In your case, the paint layer is compromised and must be removed. The only ways I can think of to remove the paint would be mechanical and chemical. Since you cannot grind the paint off, you'd be looking for a chemical stripper, and the process is nasty and horrible work. It's to be avoided at all costs.
Have you considered other options like vinyl flooring? I used vinyl sheeting to redo my laundry floor, and it looks fantastic. If you're not a fan of vinyl sheeting, then hybrid vinyl planks do a good job of imitating real timber. There are also vinyl tiles that are easy to install.
You can paint directly over what is there, but there is a reasonable chance that it will peel off if the original paint flakes. Manufacturers don't recommend painting over degraded surfaces.
Please let me know if you have any questions. Mitchell