What are my options to replace Gnome 3's Nautilus in the least intrusive way?

You can install nemo and thunar without their recommended packages. This won't pull Cinnamon desktop or the XFCE desktop.

sudo apt --no-install-recommends install nemo

Similarly for thunar

sudo apt --no-install-recommends install thunar

caja seems to install mate-desktop even with --no-install-recommends.

Nautilus used to have many more features back in 2010 (when GNOME 2 was still around), after which they began to remove useful features in the name of simplicity.

Thanks to @ArchismanPanigrahi's answer in which he revealed that those desktop-looking packages are not actually representing and not installing the entire desktops, I felt emboldened to install both Nemo and Thunar, together with their recommended packages.

sudo apt install nemo needed only ~30MB disk space, and

sudo apt install thunar needed only ~12MB disk space.

After a full reboot, on the purple GDM login screen, in the small cogwheel icon, no additional Cinnamon deskop or XFCE desktop option had shown up. (It still lists only gnome-classic, Ubuntu, and Ubuntu on Wayland)

Of course, I started to use both file managers with 256px thumbnails.

Nemo

Good stuff about Nemo:

• "everything" can be configured, and more (there is heaps of stuff in gsettings)
• works in a sensible way, remembers which specific directories were viewed in list and which ones in thumbnail view modes and applies that view mode automatically; same goes for sorting
  • subdirectories can inherit their viewmode from their parents (excellent feature, makes things easy)

Less good stuff about Nemo:

• has a bit of un-handsome looks, especially in list view
  • it uses monochrome Adwaita icons in the sidebar
• tends to develop a very bad UI lag after some time (with i915 integrated graphics)
  • it seems to set itself up for difficulties when it tries to retrieve directory file counts anew upon every step of navigation
  • the UI lag can go so far as impacting the entire gnome-shell, pretty badly
  • as far as I can tell, the UI lag can be mitigated to some decent extent by disabling the file-count displays under directories

Thunar

Good stuff about Thunar:

• even though the visual elements seem pretty similar, it has a more harmonic appearance than Nemo
  • it uses color icons in the sidebar
• through lack of decorations, it offers a very good overview of thumbnails
• remains snappy, no UI lags

Less good stuff about Thunar:

• the preferences dialog contains only a handful of settings, and there is nothing in gsettings
• the set view mode (list vs thumbnails) and sorting order applies to every directory visited thereafter; unlike Nemo, it does not try to restore the last used settings for specific directories
• it has a strange bug where some apparently random subset of the thumbnails (mixed in the same directory) shows up smaller than the rest (impacts .jpg's and .pdf's)
  • the fix seems to be easy: Thunar's "Tumbler" thumbnailing process seems to have two redundant thumbnailer plugins, resulting in race-conditions; the fix is disabling one of these plugins.