Is Budget Audiophile an oxymoron?

No. Budget audiophiles are how most of us begin. It is a matter of objectives, learning and effort as well as investment. Carefully chosen components, setup and room treatments can achieve amazing sonic levels. Put in a thousand hours of effort and what you can achieve with a few thousand dollars is amazing. Of course not nearly the same as one can achieve with a thousand hours and many tens of thousands or a hundred thousand dollars. But that is what is addictive about the pursuit... there is no end to learning, becoming a better listener, making your system sound better... and closer to the real thing.  

In general response to the question, No, if you build your own and have the where with all, time, and patience to perfect it.

My speakers cost ~$1,500 to build using quality parts from Craigs List and eBay and with an Oppo-95 and Yamaha RX-Z9 purchased new in 2003 (~$6K total), I would put it against systems at 10+ times the cost.

https://youtu.be/HXfZSbnD4pM

My answer is no, based on how I define "audiophile." An audiophile is just someone who appreciates sound quality and enjoys putting some effort in to pursuing it. It’s not about how much excellent sound quality they can afford.

The room is half the sound.

We all start with the budget thing, a tweak here, and upgrade there, then suddenly, you have more in your system than you have in your car.  Don't worry.  You can't take it with you.  That, and I use my system a lot more than I use my car.