no, keep them trim color.

and you can paint your downspouts a color to disappear against the stone.

Ditto

These are integrated gutters yes? So we are talking about the gutter lining as well as the metal lip that wraps up over the crown moulding on the fascia. Definitely the part that wraps over the front should be the trim color, the gutter lining itself could be white or dark grey to match your roof.

Paint gutters the same color as the trim they are attached to. They should blend in to whatever they are attached to, not stand out against it. So if your trim is white, the gutters should be painted white.

Take a look at the photos of my house. My house was painted with Behr Marquee paint colors Taupe Tease for the main color; Superior Blue for the trim; and Fire Cracker for the doors. It is recommended that you choose 3 colors for your house instead of two. My hubby will eventually paint our white downspots the same color as the house.

Don’t paint them red unless you want to be known as owner of The House with the Red Gutters.

Gutters should be the same color as whatever they are attached to.

The OP's house already has several colors. She doesn't need to add any additional colors to the structure of the home.

Personally, I would leave it as is. Like suedonim75 said, the house already has several colours.

You call them "tin" but, I'll bet are "terne" (an alloy of lead & tin applied over thin steel). Before proceeding further find a local guttering company that's been in business...forever. Have them pay a visit and inspect the guttering for any needed repair. Cheaper to repair (trust me) than getting new guttering. You have a lovely stone home, do not "tart it up".