Hello I wanted to write much more yesterday but I only had 160 characters available :( I mainly take photos in nature and also of the surroundings (meadows, mountains). I will have a closer look at the different pictures on Flickr.com. It's always difficult when you enter this world as a layman... And yes, I also take sports photos because I know a person who does figure skating and so it should also be suitable in this respect.

"Never good" has to be put into perspective... I would look online at a few pictures taken with the 24-240mm. So you should think a little bit about what it is mainly used for... The speed is a drawback here and at wide angle the vignetting is quite pronounced. In addition, it is not necessarily light.
Autofocus isn't exactly top notch, so it's only conditionally useful for sports, mainly because of the camera's AF.
As for light intensity... it will be difficult to shoot indoors... here the camera will quickly set the iso to 1600+... which is not necessarily bad for the camera... if you intend to do this, you should look at pictures of the camera with ISO 1600+.
The "buy/use telephoto and standard zoom separately" is also not quite correct... unfortunately there is not much choice of telephoto with Sony.... (apart from the 70-200 f4 and 2.8) but they cost a lot more and then you need a wide angle too. I have shot a lot with Canon myself and my 24-105mm was my "travel zoom" and here you even have 240mm... if you are really looking for a travel zoom then this is the ideal one.
But just... have a look at Flickr.com and enter 24-240mm in the search... then you can see for yourself how the pictures look :)
I myself use telephoto and city zooms for shootings and where high quality is important (I have the A7R II with 42MP) but for travelling I have an Olympus OMD- E-M5 II with a travel zoom ;)... for travel pictures you don't have to zoom in to 250%.... Screen size with 24MP is perfectly sufficient. Finally, if the image composition is well done, no one will look if there is less sharpness at the edges or if the vignetting is more pronounced at 24mm than at 35mm ;)
You take the photos, not the camera/lens. So don't let yourself be influenced by such comments. There are even great photos that were taken with a mobile phone.

Classic sports photography means fast long focal length lenses, and these are expensive, and there aren't really any for Sony FE (apart from the 70-200/2.8, which is at the very bottom end (in terms of focal length, not quality)). For hobby photographers who don't have an abundance of money, it's not really "worth it" because it quickly devours thousands of francs. if taking perfect sports photos isn't your greatest dream and money doesn't play such a role, then forget about sports photos (of course you can also take sports photos with a 24-240 and with good light and a lot of patience you can certainly take some great pictures with it). and concentrate on nature photos, i can recommend macro photography in particular, you can quickly achieve nice results without a lot of effort (a tripod is an advantage though, i would recommend a used gitzo, i don't think there is a better price/performance ratio).
besides, if i were you, i would ask myself what you really want to photograph and why. if you want to do really good photography, then the camera is the smallest problem, if you just want to take photos for memory (this is not meant in a derogatory way at all, i find memory photos more interesting in 99% of the cases than any technically good, but impersonal photos of some advanced ameteur), then your rx100 ii is by far sufficient, for sports you could then possibly buy an rx10 if you want to take some photos. you could buy an rx10 if you want to get "closer".

Travel zooms are never really good, if I were you I would buy/use telephoto and standard zoom separately.

The Sony 24-70mm f/4 lens is extremely well rated and also has its price, almost 1000 francs. A camera with APS-C would be great. Today I held a Canon 70D (with a 50 mm lens, I think) in my hand and such a big typical SLR camera is almost too heavy for me! I wanted to spend a maximum of 2000 francs, but I won't get far that way...