Could my rim be just a little too small? I have a recurring flat problem in my rear tire.  I can never find the obvious cause and it does not always occur in the same position on the tube.  My rim is a replacement and may be the root cause.  I have noted for some time that putting a repaired tube with outer tire back on the rim is ridiculously easy.  I do not need tire levers; my fingers will do.  Indeed, it practically a finger tip job!

I believe that putting a tire back on my original rim was not so easy; that I really did need a tire lever.  But it is so long ago that I am not sure my memory is accurate and that I am not confusing my front and back tire repairs.

So my question to others is simply whether this is plausible and whether most cyclists can expect a little resistance when they try to replace a tire on its rim.
I don't believe there are any common rim diameters where the next size down would be close enough for the difference between tyre bead diameter and rim diameter not to look wrong. The tyre would blow off the rim with very little pressure.

When finishing mounting a tyre, it's common to pull the already-mounted part of the bead into the middle of the rim where the well is deepest, to give enough slack to easily get the last part of the bead over the lip of the rim.

With narrow rims and single-wall rims, the rim well may be almost flat, leaving very little slack when mounting the bead. A thick rimtape can make this even worse.

If the new rim is a wider dual-wall rim, or you now have a thinner rimtape, you may have more slack even with matching tyre and rim sizes.

Given the recurring punctures, your rimtape is possibly inadequate or missing.