You won't see L and C in the equations that describe antennas. A quarter-wavelength conductor (half of a dipole) represents 90 degrees of waveform (360 / 4). So, when voltage is highest at the feed point of one element of a dipole, it is lowest at the far end of that conductor. At that moment, though, the voltage at the feed point of the other element is lowest, meaning the voltage at its far end is highest. Current flow is maximized and we get good radiation.
Rather than thinking of antenna elements as lumped LC, it's probably wiser to see them as sections of transmission line. LC circuits are a way of synthesizing the characteristics of transmission lines with components; we're so accustomed to using them we've forgotten the relationship.
Your magnetic loop antennas are sort of a combination of transmission line and lumped-circuit elements, as are inductively or capacitively loaded dipoles. Now it doesn't hurt to bring what we know about practical LC circuits back into the discussion. Cheap crystal sets used the self-capacitance of the tuning inductor to resonate the circuit; there was no tuning cap. The elements of your dipole work similarly. Check out an antenna-modeling program to see how they break the elements down into little pieces. Each little piece could be considered as a discrete LC circuit. The entire element then is a series of LC circuits strung together. (I should say LCR, as there is always resistive loss.)
Crystal sets without tuning caps, that rely on the inductor's self-capacitance, aren't selective. That's a negative for a receiver, less so for an antenna element (assuming the lack of selectivity isn't overwhelmingly due to resistive losses). Rebuilding the set with a better inductor and adding a tuning cap can be equated in an antenna by replacing some of the inherent LC circuits making up the elements with lumped components: loading coils, capacitance hats, or both. Those lumped components will likely have higher Q than the LC circuits we imagine constituting the elements of a full-size dipole, and bandwidth decreases.
It's tempting right now to say, Oh, higher Q means better efficiency! In the case of an antenna, not necessarily. Remember, there's no free lunch. Inductors are lossy. That's why we want to put them out near the ends of the elements where current is lowest. But then we need more inductance, which may be mechanically impractical. NFL. Capacitors are theoretically lossless, the only losses being in mechanical junctions. If they're placed at the low-current tips of the elements, they should have less negative impact on performance than inductors, but they have to be physically large to do much good. NFL.