Worth the $$ to upgrade my tonearm?

My first thought is that if you’re happy with how it currently sounds, forget the arm and buy more records.

I don’t know the RB220, except that it is Rega’s cheapest offering. The old 300 was a perfectly decent performer and presumably the same is true of the current 330, but how the 220 compares.....

Anyway, instinctively I would say that any Michell deck in good nick deserves a better arm than I suspect the RB220 is. However, I hardly think you need to spend two grand in order to upgrade from the 220. If you like the Rega you might have a look at Origin Live’s Rega-based arms. The Silver for instance is a real classic and much lauded at a price of £675 (inc. VAT) and would certainly be a significant upgrade to your current arm. It would also leave more than half your stated budget for more records.

The Silver would be my second thought then, but there are myriad others, both new and old. There's no doubt that significant sonic gains can be had with a better arm and a better cartridge, but whether or not you’ll actually hear a significant difference - well, that’s impossible to say.

+ for the Origin

And I bet your system sounds lovely !!!!

you might try some NOS tubes, small signal stuff if you have not already

To answer your question, the answer is yes, a better tonearm can make a difference.  Sometimes the improvements are small, and sometimes they are not so small.  
I recently upgraded my SME IV arm to a Tri-Planar arm, and the improvements were significant.  So much better that I am a true believer in the significance of how a tonearm affects the sound.  
I found the Tr-Planar locally as NOS for $2K.  It's not easy to find this arm so cheaply, so I got lucky.  But chances are if you are patient you should be able to find something in your price range that will make a big improvement over what you have now.  

get an SME before they are gone.  

Look at an Audiomods arm which is based on a Rega arms. Not sure of your options of adjustment on the Michell (Gyro) table but a Jelco 550 will be a solid improvement and an 850 a substantial upgrade. Not too expensive from Gem Dandy.

Origin Live Silver... 

+1 Jelco 850!

I changed from a Rega 300 to a Victor UA-7045 a few years ago & it was a nice upgrade. I have since changed arms a couple times. They do make a difference. You can still find the Victor's relatively cheap. 

+1 on the Origin Live Silver.  Excellent arm and price.

If you're dead set on upgrading your arm then +1 Origin Live. Wonderful arms.

Me, I'd be looking at a Soundsmith MI cart. 

Guys, I like the Jelco arms but they are too heavy for the 2M series of cartridges which are on the higher compliance side of things. 
A lighter arm also gives you more flexibility in the future as you can always add weight. Harder to take it away. The Origin Live arms would be a better match for his cartridge. If Paul were into Moving Coil cartridges than either arm will do. 
Don't listen to millercarbon. He thinks all of our styluses are "jittering" around in the groove. I'm all for lower effective mass but this jitter thing is nonsense. I would love to see proof that a Sound Smith cartridge tracks better than say an Ortofon at the same price point. 

I changed from a Rega 300 to a Victor UA-7045 a few years ago & it was a nice upgrade. I have since changed arms a couple times. They do make a difference. You can still find the Victor's relatively cheap.

indeed, one of the best tonearms for the money today, so much cheaper than many inferior and more expensive new tonearms, the Victor UA-7045 is fantastic if you can find a perfect sample under $1k, i managed to find NOS in the box which is more expensive, but i like NOS. The used samples are normally under $750. Now using long version of this arm UA-7082 and love it! Just make sure the counterweight is not sag down in this position. 

Regarding the quality you can easily upgrade your cartridge to another MM like Stanton CS-100 WOS, Audio-Technica AT-ML170, Grace LEVEL II ...  and the difference can be huge on your Rega arm!

But within $2k you can always buy absolutely fantastic vintage tonearms such as Technics EPA-100, Fidelity-Research FR-64fx, on the cheaper side something like Lustre GST-801 ... It is impossible that any of these arms can be beaten by Jelco, Origin Live, Rega and many modern tonearms.

If you want something special among the new ones look for a short Reed 3P

P.S. I think the weak point is your cartridge, probably it is more important to change it first. 

You can also upgrade the Michell with mods from True point Audio.
Pete's Pylons and acrylic arm-boards.