The European Championships used to have a 3rd-place playoff, but it was discontinued after Euro 1980. Attendance was poor (24,652 spectators in a stadium that could hold 81,000), and according to The Athletic, TV viewing figures were also disappointing. As a result, UEFA decided there simply wasn't enough interest in the playoff to justify continuing to hold it.
It should be noted that attendance was poor throughout the tournament, but host nation Italy (one of the teams in the 3rd-place playoff) had consistently attracted crowds of 40-60,000 during the group stages. For their third-place playoff match to experience such a massive drop in attendance convinced UEFA that people simply didn't care about its outcome.
By comparison, the third-place playoff at the 1978 World Cup attracted 69,659 spectators to a 74,624-capacity stadium, only 2,000 less than the attendance for the final. As a larger and more prestigious tournament, the World Cup is able to generate a much higher level of interest in its third-place playoff, and so has never seen fit to get rid of it.