Looks like a fairly simple Corliss valve engine. If you are refering to the big disk in the middle with the links to it, we would call that a wrist plate in the UK, and I am fairly sure that is what it is called in the original patent documentation. I can't help with dismantling methods for your wrist plate, but I would be very surprised if it was rivetted up in such a way that you could not remove it without cutting out the rivets. That would be a very poor design in my opinion. Engines such as this were usually designed so that assembly is simple and reversible without the need for special tools, although a crane and a lot of muscle may be required.
In the museum where I work we have three mill engines with Corliss valves. All 20 odd of our mill engines are kept in full running condition and we run on steam periodically during the year. One of our Corliss valved engines(a tandem compound) had a lot of wear in the bores of its hp inlet valves, to the point that they passed so much steam that the governor could not reliably control the engine. As you will know this is not a good condition to operate the engine. We have no machine tools large enough to take the whole of the hp cylinder for valve re-boring, nor could we find anyone with (say) a really big horizonal borer to take on the job. In the end we built a line boring machine around the engine and rebored both the inlet valves in situ - a long job. Of course the valve diameter didn't match the bore diameter after this, so we had to correct this by offset turning the whole valve body. Fortunately we have lathes big enough for this job.
The moral of the story - Corliss valve engines can suffer from wear which has to be corrected for safe operation, and if you have wobble on the wrist plate, I would also look at the valves themselves while you are at it.