Hi,
You may get a very small amount of water moving in the rad if the thermostat is fitted with a bleed hole ( to allow air out of the cylinder head when filling up) there should be a jiggle pin in the hole which seals it when the force of water from the pump lifts it up. This may only happen at reasonable engine speed. Although it wouldn’t make much difference to warm up time or final temp make sure there isn’t any sealer sticking the pin open.
It may be possible that the thermostat has not sealed against the top of the cylinder head, it is only clamped there lightly by the top cover. Adding an extra gasket (if the flange leaked) may allow water round the outside of the stat. See if you can move it side ways when the cover is tight ( through the top hose opening).
Sometimes these jobs never go easy, thank your stars it’s easy to get to, not like some of the modern euro boxes.
Good luck
John f