The best way to find out is to put your E61 away for a couple weeks and replace it with your trusty old Gaggia to see what you can do with it. Memories of your upgrade path may not be as delicious as you think.

I believe most of the limitation on these machines are fundamental things like brew temperature and pressure. If you can figure out how to get these to behave reasonable and reliable than you have something to work with. The much beloved Gaggia has a boiler too small for a full double without the temps dropping significantly during the shot. I think tuning the pressure down for a single or lower volume double is going in the right direction. Silvia has the offset group to complicate temperature management. I know Jim Gallt has data to show how to get the shot temps to be reasonably flat. This requires the grouphead temp be managed as well. With data I've seen, it would be easy enough to miss this mark here. Pulling shots back to back or steaming is bound to cause the group to overheat relative to the boiler temp. Sit and wait for it to cool down or come up with another strategy to deal with the problem. I know temperature is not THAT important, but I'm not talking about a couple degrees off here, much more outside an acceptable range.
These things led me to choose an Isomac Venus as a simple solution single boiler machine. It still needs the basic temp and pressure issues addressed, but a saturated group and reasonable size boiler makes it easier to manage. About the only real Venus SNAFU is they put the OPV on the wrong side of the boiler, but this can be addressed easier than some other problems.

I agree with the gaggias boiler being small for normal size doubles.. I see the same/similar issue when pulling tight triple 'strettos.

My best results with my Classic, have been using the stock faema basket which came with Anita(or, simply the stock double which came with the Classic) and pulling the shots on the tighter side, but still keeping the shot time at 30secs max. Generally, Id shoot for a 1.25/1.5oz shot, but around 27ish seconds. On anita, I pull the shot longer, but still under 35s, for the same volume(if the blend/bean calls for that, of course)

My favorite blend to use with my Classic is Klatch House espresso. For whatever reason, on Anita, I just dont enjoy the house espresso, usually opting for the 08' Comp Blend when I buy from Klatch. But I get great, chocolaty, caramely, nutty, not to mention, thick shots on the Classic. I don't use the machine very often, mostly on Vacation/travel.

There is no doubt you'll have a much easier time with a K10 and a BrewtusIII in regards to pulling shots just how you want, but the Classic is plenty capable. Im thinking that, just like with HX and DB machines, certain blends work better with the BIII then the Gaggia, and Vice-Versa.