On the other side of the kitchen wall is the bathroom sink.

This is the plumbing for my bathroom sink (photo below was taken when I re-did my bath).

Does the plumbing on this side make any difference in terms of my hooking up the washer to the kitchen sink plumbing?

You probably won't like this, but I'll give you the con.
Washer plumbing discharge are required to be 2" because 1-1/2" often can be compromised, by volume and there are specific requirements for traps and height of standpipes off the floor and above the trap.

Are you above another residence?

Discretion, being the better part of valor, I would bite the aesthetic bullet, place the washer closest to the sink and purchase the adaptor for connecting the water to the faucet and drape the discharge across the short counter to the sink, which then affords you an accumulation vessel.

You can safely purchase fittings and valves which will allow you to connect the water supply on a permanent basis, under the sink, through the cabinet.

You can also connect the discharge to the sink waste, however, probably not by anyone bonded and insured.

If you "elect" to pursue this, insure that there are NO horizontal threaded fittings.

And be prepared to be running between the kitchen and bathroom to make sure that when the washer is on discharge, those vessels are not being overtopped.

Your lingo presumes that I know the basics in terms of plumbing but the truth is that I don't.

In basic terms, are you saying that there is a way that I could connect my washer to the sink drain or not?

Or are you saying that I should dump the discharge from the washer into my kitchen sink?

To me, this sounds like you have a solution that does not entail dumping the discharge into the kitchen sink. If so, I'm VERY interested in understanding what you mean.

The photo below shows the main drain in my condo. It's across the bathroom to the corner near the toilet:

The photo shows the floor of my walk-in shower when it was being built. The PVC pipe leads to the main drain.

To access the main drain now, I would have to make the hole in the photo below bigger:

The drain is to the left of the electric cable. But you can also notice that there's a chimney in the hole as well which makes things quite tight. On the other side of the wall on the left is my walk-in shower. I don't know if a plumber can access the main drain without cutting into my shower.

One suggestion I've gotten is to draw a path from the spot in bathroom that is immediately across the wall from the spot in the kitchen where I intend to place the washer, to the main drain on the other side of the bathroom. Then to pull out the tiles on the pathway and lay down PVC pipping to the main drain. But this assumes that the plumber will be able to access the main drain from the hole near the toilet (even if it's enlarged). After the plumbing is done, I can replace the uprooted tiles. This may end up being my only alternative. Maybe.

If you could compose a floor plan showing the bathroom fixture locations as well as the kitchen sink and post it as a picture, it would be most helpful.

You can't do it legally without getting into the interior walls of the condo, and that will require the condo's permission and will not be cheap. You can do a "portable" type model that is small, inefficient, but still better than trying to wash your clothes in the tub. It would be a temp type hookup, with an adapter for the kitchen faucet and dumping into the sink. If you can't deal with that, then the laundromat is your best bet.