I voted "No," it might be useful for the tinkerer or the pro though.
If there were a third option like "Yes" "No" and "Mixed," I would have voted "Mixed."
Accessing terminal on the fly can already be done with F12 (my kids show me that) or ALT+CTRL+T.
This where you do not want it, although they are not sudo, they can do things out of control, just because they are kids.
For example: I have already erased their account then re-create it (2 times) as they have fulfilled the SSD. Don't ask me how they did it; I still do not know how they have done that, fortunately, I have created an account for each of my kids, so it was straightforward to solve the problem.
Giving my kids one more option to access terminal, can be excellent later when they will be more thinking about what they are doing instead of systematically experiencing like they are doing now.
Don't forget that I see Ubuntu-MATE as a TRUE family friendly OS, certainly, the friendliest ever made (just my opinion).
On the over hand opening Caja, navigating in folders then terminal in Caja with terminal "already" looking in the folder (so no need "cd folder") is helpful.
Maybe a compromise? for example, To make the button "Show" appear in Caja, it would be nice if it needs to be activated via the menu "View" then select "Show Terminal" or use the shortcut keyboard, but the button "show" should not shows-up by default.