@teerdurchzogen I have looked more deeply into your tournament hypothesis.
The two most popular time controls in terms of total play time are 10+0 and 3+0 and these have indeed tournaments and also a quick search button for the lobby.
However, the 3rd most popular time control 7+0 has neither tournaments nor a quick search button.
The 4th most popular time control 15+15 has no tournaments, but a quick search button. a tournament would take 3 h.
The 5th most popular time control 5+3 has no tournaments but it has a quick search button.
The 6th most popular 3+2 has both tournaments and quick search.
The 7th most popular 15+0 has neither tournaments nor quick search.
The 8th most popular 1+0 has both tournaments and quick search.
The 9th most popular 2+1 has no tournaments, but a quick search button.
The 10th most popular 5+0 has both tournaments and quick search.
So, no, I do not think tournaments or quick search buttons are decisive in time control preference.
Data processed from source above:
min sec games tot FIDE lichess play time tournament quick search
10 0 1567951 10 blitz classical 15679510 yes yes
3 0 2158945 3 blitz blitz 6476835 yes yes
7 0 588878 7 blitz blitz 4122146 no no
15 15 105757 30 rapid classical 3172710 no yes
5 3 339861 8 blitz blitz 2718888 no yes
3 2 467778 5 blitz blitz 2338890 yes yes
15 0 148208 15 rapid classical 2223120 no no
1 0 2068329 1 blitz bullet 2068329 yes yes
2 1 399411 3 blitz bullet 1198233 no yes
5 0 232056 5 blitz blitz 1160280 yes yes
5 8 86814 13 blitz classical 1128582 no no
5 5 100722 10 blitz classical 1007220 no no
20 0 42818 20 rapid classical 856360 no no
8 0 94695 8 blitz blitz 757560 no no
10 5 44987 15 rapid classical 674805 no no
2 0 233300 2 blitz bullet 466600 no no
5 2 50831 7 blitz blitz 355817 no no
5 4 35938 9 blitz classical 323442 no no
4 0 57058 4 blitz blitz 228232 no no
0.5 0 384998 0.5 blitz bullet 192499 no no
3 1 42254 4 blitz blitz 169016 no no
1 2 35681 3 blitz bullet 107043 no no
1 1 46661 2 blitz bullet 93322 no no
0.25 0 256592 0.25 blitz bullet 64148 no no
0 1 58782 1 blitz bullet 58782 no no