Air carrier Captain reported ATC issued a departure heading from TYS that the flight could not comply with. Captain recommended ATC departure instructions should be consistent with the ODP or to consider creating a new SID.

Narrative

Knoxville was departing runway 23R with 23L closed. We received a verbal clearance of Cleared as Filed with no departure instructions at all. FO programmed FMS initially and was to be Pilot Flying this leg. During the route check; he informed me there was an ODP (Obstacle Departure Procedure) and he had added it to the FMS; as there was no departure instructions given (verbal clearance received). Route was checked and verified for departure via ODP and flight planned route. It was noted that terrain was a threat; with a minimum safe altitude of 5100 for direction of flight; up to 7900' to south east of course; field elevation 953'. I pointed out to FO that once we had radar contact" and were given instructions we could proceed to first fix if we had not completed the full ODP. ODP for 23R was "Climb on VXV VOR R-226 to 4300' before proceeding on course". This ODP roughly corresponds to runway heading to 4300'. I checked to make there was no DVA (Diverse Vector Area) also published; and FO as PF briefed the departure. Taxi to runway was uneventful but busy due to MEL on APU and second engine start. We waited at hold line for a substantial time period due to an unusual heavy amount of arrival traffic. Finally; tower cleared us to line up and wait due to roll out traffic; but warned us of traffic on final about 2.5 miles. Did I mention it was busy; most likely due to closed runway. I proceeded on to runway with throttles up. Halfway through turn to centerline; we received "Cleared for takeoff; fly heading 250". I stated "We cant do that" and FO transmitted immediately "We need to fly the ODP". Tower came back with "OK; then fly runway heading". This was acceptable as it was consistent with the ODP. We were rolling and I was aware of landing traffic 1.5 to 2 miles now behind us; so I TOGA and proceeded on takeoff. I totally forgot to transfer controls to FO for his takeoff. Noted error and told him I would continue for flight safety. Takeoff was uneventful. At approximately 1500 AGL we were given a frequency change; making contact at about 2000' AGL. After receiving "radar contact" and permission to turn on course direct first fix; we turned on course; and I transferred control back to FO to serve as PF for rest of leg. Radios were quite quiet; so I asked departure control if the tower normally gave headings on departure and was departure aware? Departure Controller stated that their LOA allowed tower to issue heading as they see fit. They; Departure Control; were not necessarily aware ahead of time. Aware of our request earlier; he stated "We don't issue the SID (KNOXVILLE7) but everyone flies it anyway as if we did". Both FO and I were shocked he stated this over radio. They are issuing headings off the ground with no authorization; assuming that everyone is following a SID they did not issue and most likely is not valid after looking at it. The KNOXVILLE7 is an old radar vector SID that is not current with appropriate criteria for its instructions. For it to be valid; airport would need a DVA I think which is probably the real reason they do not issue it.Recommend notifying TYS that the only valid IFR departure they can issue right now is "fly on course"; the ODP; or issue "Fly runway Heading" which is consistent with the ODP in anticipation of radar contact (per 7110.65) until they get a SID they can issue or a DVA. Since they seem happy with status quo; I would recommend a DVA; which would be within criteria of 7210.3 for their facility if they do not have a SID they can issue; and probably quickest; easiest solution."