Then there's no reason why non-Catholics cannot perform successful exorcisms, then? Protestants, new-agers, etc.?

You're missing the difference between a real possession, which requires an exorcism from a priest, and lesser forms of demonic activity, i.e. curses, oppression.  These lesser forms of demonic activity do NOT require an exorcism and the Church usually does not allow for one.  These lessor forms can be reduced or eliminated in a variety of ways (i.e. confession, prayers, holy water, etc).  These lessor forms of demonic activity could be eliminated by prayers from the laity or even non-catholics.  Prayer is powerful.

Lots of possibilities here --

1) OP's condition was not one of true possession.

2) There's nothing to prevent God from listening to private prayers and driving the demon out.  God can do whatever He wills.  

3) In fact, exorcism isn't primarily a power of order, but rather a power of authority or jurisdiction.  So if the man had been acting from the authorization of the Church (for those of us who believe that it remains even materially), the authority of the Church through the man would suffice.  Exorcist is not a major order and does not involve some kind of character in the man who receives it.  It is an order received prior to the Major Orders and involves a participation in the powers of the priesthood without having the priestly character.  In practice, the Church rarely authorizes someone who isn't a priest to perform exorcisms, but there's nothing to prevent the Church from authorizing a layman even from exercising authority ... conferring ad hoc as it were the Minor Order of exorcist.  Many theologians consider the Minor Orders to be functional (through authority) rather than an actual participation in the priesthood, in other words, receive some of the authority of a priest without actually being a priest.

4) Demon could have left of its own volition PRECISELY hoping for the effect of confusing the OP (and others with whom OP might share the story) about the status of the Novus Ordo.

Just like all those cases of Novus Ordo "Eurcharistic" "miracles".  These prove nothing.  It's possible for the devil to fake such things precisely in order to confuse people about the validity of the New Mass.

Catholics do not form theological opinions based on observed phenomena like this.  It's the other way around.  In fact, that's exactly the mind of the Church.  When such phenomena take place, the FIRST thing the Church does is to examine it from the theological perspective.  If there's bad theology or other aspects not in keeping with the Catholic faith, the Church concludes from those things that the phenomena are not of God.  In other words, the Church does not embrace the false theology BECAUSE of the phenomena, but, rather, rejects the phenomena in the face of bad theology.  It is PRECISELY the intent of the devil in perpetrating such phenomena to cause people to embrace a false theology or false spirituality.

So, in the end, it is dangerous to conclude from OP's experiences that the Novus Ordo orders are necessarily valid.  If I were the devil and knew them to be invalid, then what better plan would there be than to fake successful exorcisms, Eucharistic miracles, or other types of miracles to deceive people into accepting them as valid?

Here's how I reason about the Argentinian so-called "Eucharistic" "miracles".  I have concluded from Catholic principles that the New Mass displeases God at the very least and is quite possibly invalid (or doubtfully valid).  I do not believe that God would perform a miracle that might endorse or otherwise encourage participation in this Mass, since it displeases Him.  Consequently, I believe those purported miracles to be false ... or at the very least doubtful (since the conclusions are my own and not direct teaching of the Church).  I DO NOT take those "miracles" and then suddenly conclude that the New Mass is just wonderful and pleasing to God and/or certainly valid.  But some Trads argue (falsely) from those phenomena that the New Mass is valid.  Inconclusive at best.

I would agree that those are all possibilities.  But it makes things rather difficult, because to remain consistent, one has to pretty much abandon that there is anything significant about traditionalist exorcisms since in those cases the same possibilities would hold.  "Sometimes demons just leave." 

That's some despairing logic.  History shows that sometimes many, many exorcisms are required for certain cases.  The Church has never claimed that an exorcism works 100% of the time, every time.  She only claims that She and the priesthood have power over the devils, through Christ.  Ultimately, it is God (and the afflicted person) who decides when/if a possession will end.  The Church and the priest are merely instruments to show God's power.