Air taxi Captain reported an overshoot of the ASE LOC/DME-E final approach course; citing strong winds and tight vectors as contributing. Reportedly; there have recently been other flights involving overshoots and descent below published altitudes on the LOC-DME approach.

Narrative

We were given very tight vectors from the west to join the LOC-DME/E at ASE; with a strong wind from the west; causing us to overshoot final. The autopilot was correcting as were were descending; but I picked up and verified ASE runway and decided to proceed visually as we were VMC and clear of clouds. I continued to correct back to final but the wind made that very challenging. I proceeded visually to descend and line up on final. In the heat of the moment; I missed whether we were updated to cleared visual approach; but continued to descend and line up on final due to the critical nature of descending to prevent a very steep approach at ASE. We were VMC the whole way and correcting to final. The approach was controlled and very normal for an approach into ASE. Once we landed; we were asked to call tower to discuss the event. They relayed to us that they had had a rash of overshooting finals lately and that pilots were descending below published altitudes. For us; the problem started with the tight vectors to the overshoot with the west wind complicating getting back to final. Once below the clouds; I descended to prevent an overly steep approach; which is what you get if you fly it as published. In the heat of the moment correcting back to center and descending; I missed whether we had been updated from cleared the LOC-DME/E to a visual approach. In the future; I will make sure we have been cleared to proceed visually before descending to prevent the steep approach.