What I called ridiculous was not your overall objections, but your specific claim that the tower would have to be hundreds of kilometers wide at the base. I’ve no idea what you have in mind as a design, but whatever it is, you’ve done something wrong if it has to be that wide.
Regarding the footings, a 1-m slice of the tower would weigh in the ballpark of 10^6 kg. Which means the average ground pressure is a measly 1 kPa. If we limit ourselves to ground loadings of 100 MPa, it means we’re only spending 0.001% of our ground area on footings. It’s so little that we can afford to spend a lot on seismically tolerant footings that don’t couple ground movement into the structure.
Wind is obviously a consideration, but less so at the higher altitudes where the air pressure is almost nothing. The upper parts of the structure can be relatively flimsy; at the top, the pressure is roughly that of Mars. Truss structures obviously do well overall when it comes to wind due to their low cross-sectional area (compared to solid structures of similar size), particularly if the components are reasonably aerodynamic.
The structure would obviously have many damping elements to remove internal energies, probably in the form of linear shock absorbers and damped masses. Allowing resonant modes to propagate unconstrained throughout the structure is a non-starter. Active elements are a possibility but probably overkill. Large damping elements are in widespread use in bridges and the like.
Redundancy is a potential problem with trusses as they are not always immediately amenable to simply adding more members, and balancing load between members in the normal case can be challenging–it can be easy for all of the load to be handled by just one supporting member. It helps if the design is not too rigid. This aspect probably requires more research but is hardly a project-killer.
While a 30-km version of the Gateway Arch would certainly look cool (I can only assume you had something like this in mind when you say “catenary supported”), it seems like wild overkill. If nothing else, you need a temporary structure to hold it up during construction. That temporary structure is… very likely to be some kind of truss. Since this isn’t supposed to be pretty, I see no reason not to just stick with the truss. A solid arch has ground pressure problems as well, unlike the truss where the load can be spread out over a large area.
Using tensile fiber elements (probably in the form of pressurized cylinders) would certainly make the structure much lighter, but is outside the range of widely-used and understood designs.