I agree, the first thing I would do would be to change the backsplash to something light and simple

Replace the backsplash.

Maybe a lighter counter? If you're going to change the backsplash (which I agree with others about), decide if you're going to change the counters because they should be changed first and will be a key driver when selecting a new backsplash.

The counters look like granite so a fair expence to replace. Perhaps a first step could be removing the tile, repairing the dry wall, and then just painting the dry wall to see the overall effect. Doing that, will really tell you if the counters are still an issue.

You could also just paint foam board and cover the tiled backsplash in a prominent area of the kitchen to get a truer sense of the counter without it fighting the backsplash.

I saw painted over backsplash that looked good. It is the least expensive quick fix ! And if you hate it, you were going to tear it out anyway. Google painting backsplash tile or something like that.

Agree with others, remove backsplash. That will make a huge improvement.

Since white kitchens is your preference. I would hold on new countertop.

If you do decide to paint cabinets, get a professional cabinet refacer. Wait 3+ months to verify the paint job is holding up to your liking before putting $$$ countertop in.

There's too many times painting cabinets which had unforseen problems holding up. It would be a real expensive disappointment to put $$$ countertop on cabinets that you don't love.

Once base cabinets are to your liking, treat yourself to a beautiful countertop.