Yes, remove the cabinets over the countertop.

So you don't lose any storage space, I would re mount them on the outside wall if they will fit that space.

to match the existing fronts, or the stained glass side if you like them... I'd remove the stained glass panels and either use the glass panels in a different application somehwere else in the home, or sell them on marketplace... someone out there loves an Iris stained glass panel.

remove only the last cabinet.... [near the open end of peninsula].....that will do a lot for making it less obtrusive and you get to keep the fabulous glass. look for a place to use either the stained glass section for display....or the entire cabinet that you remove..somewhere else.

Looks like you're not planning on removing the cabinets but only dealing with the stained glass. I suggest you replace the stained glass with sanded glass (or plain glass that you can cover with a sanded glass look film). I would also suggest you recover your stools in a lighter fabric colour for contrast.

Do you need the storage space? If not, then I vote in favor of removing the cabinets. It would make the kitchen feel larger to eliminate that visual barrier. If keeping the storage is necessary, then I like @lisedv 's idea of simpler glass. Maybe even additional glass doors on the kitchen side, to allow some light to come through.

This is akitchen I did for a client just look at the change when the hanging cabinets were removed nothing about the size of the space changed

If you remove the cabinets, there may/may not be a need for patching of the stucco/popcorn finish ceiling underneath them.

I removed the cabinets over my peninsula in my last home. It was the best improvement I made to that house. Opened the kitchen to the family room and gave us a functional peninsula for serving buffets and/or eating meals.

You may want to just remove the 3 cabinets furthest from the wall. The one attached to the wall can still be useful storage and isn't really impacting the view since it is in line with the other cabinets on that wall.

On a side note - I highly recommend recoloring the grout on your tile floor. It is an easy to accomplish DIY project that can really make a difference.

I would consider the following:

1. Remove the three peninsula cabinets furthest from the wall.
2. Rotate the remaining peninsula cabinet 90 degree so its door faces the same way as the cabinets to its left.
3. Replace that cabinet's door with the non-stained-glass door from the tall cabinet you removed so it matches the doors of the adjacent cabinets.