I'll leave the subject of manuals to Bruce. Leaded fuel car engines did rely on the lead, to some extent to lubricate the valve stems, but more importantly to lubricate the valve seats and prevent valve burning. However mower engines just don't last long enough for this to be an issue. A car engine lasts - before overhaul - 100,000 miles at an average speed of less than 30 mph, or more than 3,000 hours. Relatively few push-mowers last 100 hours. These engines have aluminium bores and aren't a long-life proposition. The lack of lead just isn't a problem.
My solution to the trash in the fuel tank problem is to drain the tank, removing the trash along with the fuel. If any remains, just let the tank dry out and the remaining trash will become loose powder that can be wiped out. I wouldn't disturb the fuel pump diaphragm (between the tank and carburetor) unless it becomes necessary, because you will almost certainly then need a new diaphragm in view of the engine's age.
