A Quality experienced service plumber can and they frequently can get the drain tile flowing again. It is not cheap but the alternative is to waterproof the entire basement. When the young bucks say it can't be done what they mean is that they don't know how to do it
If the weeping tile system is literally made of tiles (short segments of usually ceramic pipes each wider at one end so they fit together loosely) then a snake with cutting head may have difficulty going through because the inside diameter keeps changing.
When cleaning anything including baby diapers, it is better to sweep off or scrape off as much solid and semisolid material as you can first. (1) Less cleaning fluid or detergent needs to be used, (2) Less of a mushy mess which in turn means that bleaches or disinfectants will be more effective.