Is "recent past" an oxymoron?

No. You are correct. It is both common and acceptable to qualify the past as the distant past or the recent past.

You are correct about usage: it is not an oxymoron, because "recent" and "past" are not in opposition. The word "past" in that phrase is, to an extent, redundant. An oxymoron would be an instance of two words whose meanings suggest each other's opposite, or at least significantly dissimilar concepts. As a comment suggests, a phrase containing some redundancy such as "recent past" might be called a pleonasm, though I don't think it's a common term in regular conversation.

I cannot see why it is wrong to modify the noun past with the adjective recent. It conveys to me the idea of a time neither recent nor in the remote or undetermined past but somewhere in between.

"recent past" as compared to "distant past".

Both are "past", but "recent" is closer to present than "distant". So both "recent" and "distant" are valid qualifiers for "past", so not an oxymoron.