I am in VMware Workstation (networking very similar to Player) and I am in a Kali 2021.1 machine here (I have about a dozen machines in this computer).
Host Only is disconnected from the outside world and is more for isolated testing. It is not used for general work where Internet is required.
NAT / Bridged both allow you to use Internet freely. Bridged networking uses DHCP from your router outside your Host machine. NAT networking use VMware's NAT engine and provides DHCP to the guest machines in a different subnet than the host.
You need to be aware of which networking you are using as both work. Bridged networking is on the same subnet as your router and this can facilitate file sharing across machines.
NAT / Bridged is not a "safety" issue any more than a Host machine (which is likely where you have your email and stuff). Host Only is "safe" only because it is very restricted. It is not helpful for general work.
I do not use and cannot say about Kioptrix, but both Kali and Ubuntu (and all Windows) work fine with either NAT or Bridged.
Kali can only detect Kioptrix when two VMs both under bridged mode.
That is likely due to Kiptrix more that Kali. I can share files Kali to Host and I can certainly connect to Ubuntu and this is with Kali and Ubuntu using NAT networking on my Windows 10 host computer. So it appears the issue you are having is due to Kioptrix or a compatibility issue between Kali and Kioptrix.
I can setup metaexploitable VM properly, and Kali can detect it.
That seems normal.
Can anyone explain to me what could cause the automatic mode switch?
I do not see this at all - multiple VM's and 3 running here all simultaneously. No mode switches at all.
See if you have in your Programs, VMware, Virtual Network Editor. Launch the Network Editor Go to the NAT setup and Reset NAT. This will change your NAT Subnet and requires you to restart the computer. At this time, it would be useful to Repair VMware Player by reinstalling it (no impact on machines).
Now let's deal with your machines. Turn off Kioptrix and leave it off. Update Kali and then shut it down.
Now restart your host system, start VMware Player and launch just Kali. It should launch properly and not change Modes.
Note: The Kioptrix web page says DHCP is enabled within Kioptrix and that IP address is automatic. You probably want to turn that OFF.
This should provide some good information to get you started. I have been using VMware Workstation since VMware Version 1 on Windows NT 4 Workstation and all versions up to V 16 on Windows 10. So I am familiar with using NAT Networking on Laptops (not always connected to the outside world) and Bridged Networking on my Desktop computers (always connected).