D01 TRACON Controller reported issues with a MEDEVAC aircraft not being able to get an expedited routing and becoming a conflict with another aircraft.

Narrative

I am reporting this event after it was brought to my attention by the controller working the sector. I reviewed and derived details of the event from the FALCON replay. XA:11Z: ZDV apreqs Aircraft X direct to APA as 'Critical' which would have the aircraft track over SOLAR on the D01/ZDV boundary. No altitude was specified. The DR4 controller accepts the apreq. This aircraft should have originally be assigned the ZOMBZ3 arrival. XA:21Z: DR4 conducts a position relief briefing without mentioning that an aircraft had been approved direct APA.XA:30Z: ZDV Sector 41 initiates a hand off of Aircraft X roughly 10 miles south of SOLAR descending out of FL240 to 17;000. ZDV41 calls DR4 and states that they weren't sure what was coordinated as the coordination was roughly 30 minutes prior; they also request control on Aircraft Y. Aircraft Y had been a departure from APA pointed out from SR4 to DR4 to climb to FL200 for the ALS transition; which routes over SOLAR southbound. This aircraft was in direct conflict with Aircraft X as Aircraft X was assigned 17;000 from ZDV without coordination. DR4 stops Aircraft X at FL210 as soon as the pilot checked in. No LoSS occurred.DR has to then vector and descend Aircraft X through several DEN departures well east of APA and sequence Aircraft X behind slower turboprop traffic because that was the only place to effectively fit them into the flow of APA traffic. Had Aircraft X been routed via the ZOMBZ3 arrival; they would have been de-conflicted with Aircraft Y and all of the DEN traffic. Aircraft X would have likely been cleared direct to APA within D01 airspace and expedited to the airfield because there was less traffic west of APA. We are having a significant issue between ZDV and D01 regarding priority handling of MEDEVAC aircraft. Situations like this highlight exactly the reasons we have tried to explain to ZDV that simply direct destination is NOT advantageous to all aircraft. In fact; it often times delays MEDEVAC aircraft even more and is clearly unsafe. We have discussed this issue with NATCA and Management at ZDV several times now; however they refuse to comply with the LOA because their interpretation of priority handling is not consistent with reality of the NAS in general; and expeditious terminal operations.