The issue was caused by the vmnet.ko kernel module not being loaded automatically at boot, which resulted in networking not working and the "Connected" checkbox being grayed out in VMware.
So...
Rebuilt and installed VMware kernel modules manually:
sudo vmware-modconfig --console --install-all
Verified vmnet.ko exists:
sudo find /lib/modules/$(uname -r) -name vmnet.ko
Manually loaded it:
sudo modprobe /lib/modules/$(uname -r)/misc/vmnet.ko
Confirmed it worked:
"Connected" became active, and lsmod | grep vmnet showed the module loaded.
Made it load automatically at boot:
echo vmnet | sudo tee /etc/modules-load.d/vmware.conf
Now everything works fine even after reboot.