Flight Instruction with student reported during arrival they descended below the Minimum Vectoring Altitude by more than 500 feet and believes they should have cancelled their IFR clearance sooner and continued with a visual approach to avoid the altitude deviation.

Narrative

Vectors to left downwind Runway XX; off flight plan approach. Descend from 13;000 then quick succession of descent to lower altitudes: 12; 9; 8.4. By that time the PIC was already on base for [Runway] XX and way to high. Light to moderate turbulence while passing over a mountain and wind sheer warning additional moderate turbulence on base just after answering ZZZ Approach inquiry to how our flight was going with relation to turbulence. Just after explaining we were experiencing mod turbulence we experienced more trying to turning to final. Multiple planes inbound for [Runway] YY and we seemed to be transferred late to ZZZ tower in a last minute rush; when we were actually already on short final and still 4000 feet above TDZE... We were on base when asked if we had runway in sight and then finally received our clearance to land. With incoming traffic imminent communications seemed overwhelmed or over saturated with ZZZ Approach. My student literally had to chop and drop in order to even make the field; otherwise we would've definitely had to do a go around. Again we were high when cleared for visual and expected to descend 4000 ft. on a 3 mile final. This is outside the realm of a standard approach to land in my experience. ZZZ tower only really had time once we were transferred to clear us to land. Then gave us taxi instructions to FBO. After having tied up the plane and walked in and spent 20 minutes debriefing an airport ops manager walked into the FBO and asked for me by name then said the ZZZ Approach wanted my contact information. As safety pilot for Person A I was surprised to have them looking for me because I wasn't on or apart of the flight plan at all and my guess is Person B had talked to another pilot preparing to fly and sent him to me. When I spoke with ZZZ Approach they said that there was a Minimum Vectoring Altitude issue which was set at 8.4 and we ended up below it to about 7.9; and being that it was more than 500 ft. below = 'significant event'. Having flown primarily IFR flight plans and training for the past 3 yrs I was surprised to hear about all this and in this fashion which seemed to be very non-standard and way delayed and unorganized. I get that we may have made a mistake and in retrospect knowing things were spiraling out of normal ops I should've canceled IFR in the air and continued with the Visual in VFR rules and asked for a frequency change to the tower to alleviate the red tape of IFR rules and ultimately get me into the hands of the ZZZ Tower and the people I needed to talk with. Having left Airline X in Month and moved back to ZZZ from ZZZ1 where the airspace is much busier; I'm surprised as the level of disorganization or difficulty trying to get approaches and cooperation in general. I don't mean to sound accusatory or malicious in any way but in comparison to the level of activity I'm used to I'm surprised to experience such issues. ZZZ2 was the 5 busiest airport behind Ohair multiple time in the last few years. I know I make mistakes and everyone does so there's no perfect scenario. I just thought it was worth making an official filed debrief of my experience going forward to see what I can do better as a pilot and instructor. Also so learn what I can expect in the future. What I've found in our current unique aviation environment is that there is a significant need for experienced instructors all over but especially in ZZZ currently and my schedule is getting loaded with need for anyone with experience beyond an initial CFI capability. Safety is my biggest concern as well as teaching best practices and instilling an earnest training program that will not only help my students succeed but make them better pilots for the industry so they can also hold a high standard. In accordance with my time spent instructing with flight school X and XX different airline cadet programs from all over the world as an European Union Aviation Safety Agency (EASA) authorized instructor as well asFAA. As far as I know a minimum vectoring altitude is not common pilot terminology but for ATC. That being said that we descended below it and having full awareness in VFR conditions as well as the Moderate turbulence and wind sheer warning experienced with incoming traffic for perpendicular runways I believe it was necessary for the safety of the flight in order to successfully land on the intended runway. I appreciate your time would like to connect with someone to discuss and learn from perspectives I'm not privy to in order to further develop my own capacities and pass on the knowledge wherever I can. My sincerest apologies for any headache we may have caused I've never had to contact approach in my 1300 hours of duel given with 900 of it in instrument flying. Please let me know what I can do to help.