I spun a bearing in my 427 and most everybody told me to replace the cooler - which I intended to do but ordered one with -12 instead of -10 fittings. I then took mine out and tried blowing it out and never saw anything that looked suspicious so I re-installed it. No - the
oil goes through the filter and THEN to the
oil cooler, then to the engine. What most folks would caution you about is the bypass in the filter itself. #2 - The coolers can't really be cleaned but I wouldn't worry about it.
I spent 11K total on my rebuild and rolled the dice on a $100 cooler - most probably consider me stupid but that wouldn't be the first time! Best wishes!