Unless you're being pedantic by differentiating between the correct #11 tension and the incorrect natural 11 void note, those two statements are contradictory:
If the chord symbol C13 "implies all the extensions below by default" and you explicitly don't want to hear the #11 you do have to indicate a symbol like C13(add9, no 11). Or C13(add9, no #11) if you prefer.
Chord symbols are shorthand. The problem is, as soon as you try to use them to indicate concepts that can't be easily condensed and comprehended with shorthand symbols, they fail.
fwiw one of my favorite chords is a dominant 7sus4 with the major 3rd on top, so that you get that major 7th interval between the 4(11) and the 3. Of course, technically (and sonically) the major third is the tension in that case; it's a very piquant and unique variation of the sus4 chord, not a major triad with an extra tension/extension. But if you've never heard one, check it out!