I don't know about TK but I used to run lost and found at a station and my clients included EK, BA and IB.
EK was very strict. If you did not file a DPR on arrival at the baggage hall they simply would not consider any later claim. Having said that, EK were adamant there were staff beside the carousel and in the baggage office from first bag on belt till all bags collected by pax and any unclaimed had been collected and processed in office. They also provided copious signage to instruct passengers of baggage rules in several languages. But even a F pax who paid 25k for his ticket would not be able to open a DPR after leaving the baggage hall with his baggage.
BA and IB were slightly more generous. We still had staff at carousel and in office but they might allow a passenger to open a DPR within 24 hrs of arrival. This was even more likely for premium class pax or Executive Club members.
I only have experience of losing baggage with TK as a passenger and the service was atrocious. They used a handling agent who couldn't use World Tracer. Writing to baggage tracing in IST didn't work and I had to find the local station manager and get him to personally alter the World Tracer file. 24 hours later a match was found and my bad was rushed to me. But I wouldn't expect a normal pax to have the perserverence to find a station manager or the know how to ensure file was correct on World Tracer.
I think it will be a huge hassle to get even an acknowledgement from TK that it was at fault for not having staff on hand at reclaim let alone to get them to open and process a DPR now.
Travel insurance is only real option ...