The airport has basically no real traffic aside from this Finnair flight. The other traffic is general aviation and mostly school flights of the Estonian Aviation Academy. There is no real reason to have an ILS system and tower services in place, it does not make financial sense. The charts still list ILS available for RW26 but GNSS is required for initial approach and missed approach. Without radar control to replace those parts with vectors, not too many options left. No other NPA approaches available.
In recently published maps regarding GPS interference reports, there were none over Tartu, but some just North/Northwest of Tartu, pretty much where the flight diverted. It is likely the crew encountered the interference in flight, had no RAIM prediction anymore and possibly that prohibits them from performing the RNP. I do not know the procedures at Finnair, but more and more operators prohibit visuals at night nowadays.
If I had the fuel and nothing in my manuals to prohibit it, I would have likely continued on a bit further to see if the interference passes. It has been quite localized around Estonia and Helsinki (over the Gulf of Finland, over and around Amari airbase mostly). Usually you pass through the affected area pretty quickly.