Rootless connection always set as default when removing a machine

Issue Description

In Podman Desktop we saw some users complaining about the UI not displaying resources/containers/.... and every time the problem seems to related to the default connection set.

I saw that when deleting a podman machine the remaining machine is set to use a rootless connection when it is not actually correct.

Steps to reproduce the issue

Create a rootful machine and start it
Create another rootful/rootless (it's the same) machine and start it
At this point, on Podman Desktop, you will be asked if you want to update the default connection -> yes
Stop the latest rootful/rootless machine -> on Podman Desktop you will be asked if you want to update the default connection -> yes -> the rootful connection of the first machine is set to default
delete the latest rootful/rootless machine.
if you check again you will see the default connection of the first machine has been updated to the rootless one

Describe the results you received

The rootless connection is used while i think i'm using the rootful connection. I don't see any existing resource and i think podman is not working.

Describe the results you expected

I would expect that the rootful connection is used by default for the first machine

podman info output

host:
  arch: amd64
  buildahVersion: 1.35.4
  cgroupControllers: []
  cgroupManager: cgroupfs
  cgroupVersion: v1
  conmon:
    package: conmon-2.1.10-1.fc40.x86_64
    path: /usr/bin/conmon
    version: 'conmon version 2.1.10, commit: '
  cpuUtilization:
    idlePercent: 99.48
    systemPercent: 0.29
    userPercent: 0.23
  cpus: 12
  databaseBackend: sqlite
  distribution:
    distribution: fedora
    variant: container
    version: "40"
  eventLogger: journald
  freeLocks: 2048
  hostname: LucaW11
  idMappings:
    gidmap:
    - container_id: 0
      host_id: 1000
      size: 1
    - container_id: 1
      host_id: 524288
      size: 65536
    uidmap:
    - container_id: 0
      host_id: 1000
      size: 1
    - container_id: 1
      host_id: 524288
      size: 65536
  kernel: 5.15.146.1-microsoft-standard-WSL2
  linkmode: dynamic
  logDriver: journald
  memFree: 15920971776
  memTotal: 16646393856
  networkBackend: netavark
  networkBackendInfo:
    backend: netavark
    dns:
      package: aardvark-dns-1.10.0-1.fc40.x86_64
      path: /usr/libexec/podman/aardvark-dns
      version: aardvark-dns 1.10.0
    package: netavark-1.10.3-3.fc40.x86_64
    path: /usr/libexec/podman/netavark
    version: netavark 1.10.3
  ociRuntime:
    name: crun
    package: crun-1.15-1.fc40.x86_64
    path: /usr/bin/crun
    version: |-
      crun version 1.15
      commit: e6eacaf4034e84185fd8780ac9262bbf57082278
      rundir: /run/user/1000/crun
      spec: 1.0.0
      +SYSTEMD +SELINUX +APPARMOR +CAP +SECCOMP +EBPF +CRIU +LIBKRUN +WASM:wasmedge +YAJL
  os: linux
  pasta:
    executable: /usr/bin/pasta
    package: passt-0^20240510.g7288448-1.fc40.x86_64
    version: |
      pasta 0^20240510.g7288448-1.fc40.x86_64
      Copyright Red Hat
      GNU General Public License, version 2 or later
        <https://www.gnu.org/licenses/old-licenses/gpl-2.0.html>
      This is free software: you are free to change and redistribute it.
      There is NO WARRANTY, to the extent permitted by law.
  remoteSocket:
    exists: true
    path: /run/user/1000/podman/podman.sock
  rootlessNetworkCmd: ""
  security:
    apparmorEnabled: false
    capabilities: CAP_CHOWN,CAP_DAC_OVERRIDE,CAP_FOWNER,CAP_FSETID,CAP_KILL,CAP_NET_BIND_SERVICE,CAP_SETFCAP,CAP_SETGID,CAP_SETPCAP,CAP_SETUID,CAP_SYS_CHROOT
    rootless: true
    seccompEnabled: true
    seccompProfilePath: /usr/share/containers/seccomp.json
    selinuxEnabled: false
  serviceIsRemote: true
  slirp4netns:
    executable: ""
    package: ""
    version: ""
  swapFree: 4294967296
  swapTotal: 4294967296
  uptime: 0h 23m 55.00s
  variant: ""
plugins:
  authorization: null
  log:
  - k8s-file
  - none
  - passthrough
  - journald
  network:
  - bridge
  - macvlan
  - ipvlan
  volume:
  - local
registries:
  search:
  - docker.io
store:
  configFile: /home/user/.config/containers/storage.conf
  containerStore:
    number: 0
    paused: 0
    running: 0
    stopped: 0
  graphDriverName: overlay
  graphOptions: {}
  graphRoot: /home/user/.local/share/containers/storage
  graphRootAllocated: 1081101176832
  graphRootUsed: 7895339008
  graphStatus:
    Backing Filesystem: extfs
    Native Overlay Diff: "true"
    Supports d_type: "true"
    Supports shifting: "false"
    Supports volatile: "true"
    Using metacopy: "false"
  imageCopyTmpDir: /var/tmp
  imageStore:
    number: 0
  runRoot: /run/user/1000/containers
  transientStore: false
  volumePath: /home/user/.local/share/containers/storage/volumes
version:
  APIVersion: 5.0.3
  Built: 1715299200
  BuiltTime: Fri May 10 02:00:00 2024
  GitCommit: ""
  GoVersion: go1.22.2
  Os: linux
  OsArch: linux/amd64
  Version: 5.0.3

Podman in a container

Privileged Or Rootless

None

Upstream Latest Release

Yes

Additional environment details

I'm on Windows 11

podman version
Client:       Podman Engine
Version:      5.1.1
API Version:  5.1.1
Go Version:   go1.22.3
Git Commit:   bda6eb03dcbcf12a5b7ae004c1240e38dd056d24
Built:        Tue Jun  4 23:40:05 2024
OS/Arch:      windows/amd64

Server:       Podman Engine
Version:      5.0.3
API Version:  5.0.3
Go Version:   go1.22.2
Built:        Fri May 10 02:00:00 2024
OS/Arch:      linux/amd64

Additional information

Additional information like issue happens only occasionally or issue happens with a particular architecture or on a particular setting

containers / podman