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.
Stop wasting time re-adjusting the pattern. Have several lawn sprinklers, one for each pattern.