Now that I've watched all the episodes (translated Subs, rather than the English Dub), I can give a timeline.
- Episode 165/166: Cell says he'll give the Z-Fighters 10-days to prepare for his tournament. Trunks confirms its still 10-days away in the next episode.
- Episode 167-169: Yamcha and Cell say there's 9-days until the tournament. Piccolo and Goku confirm there's 9-days in the next episode, and 169 takes place within the same day. Piccolo enters the Time Chamber on this day.
- Episode 170-173: There's 8-days in 170. The narrator says there's 8-days during the opening narration episode in 171, and Piccolo's 24-hours is up in episode 172, allowing Vegeta to enter. However, the opening narration in 171 was potentially referring to the last episode, as it's randomly Gohan's birthday, and 171 ended at night time, yet 172 opens in the day, meaning that Popo could have been talking about a second portion of Piccolo's training.
- Episode 174: 173 ended at night, while 174 takes place during the day, and Vegeta exits the Time Chamber. Gohan, having been left at Kami's Palace in the previous episode, is doing some homework from Chi-Chi, suggesting more time than one day has passed, which is supported by the Cell Games being presented as very soon in the beginning of this episode. As Goku finishes a test, an entire day passes, meaning there's at most 4 to 6-days, which makes sense with Trunks training in the Time Chamber as well. The episode ends with Goku having collected the Dragon Balls, abruptly stating that the 10-days is up.
So these episodes took up at least 5 days of time, if we assume the minimum possible span of time (Vegeta, Piccolo and Trunks only went in for one day, there's only one day between 173 and 174, etc).
The order of the Time Chamber is Piccolo, Vegeta (
all by himself), and
then Trunks. Tien refuses to go in, and it's implied that Krillin and Yamcha were chilling at Kami House. Vegeta initially says
he's going to be there for 1 day,
but says he'll take 8 instead, until Piccolo tells him he can only spend 2 in there. Trunks responds
that they can have 23 more hours than a day. Notably, Piccolo even advises Goku to spend another day in the Time Chamber. Therefore, it's highly probable that at least Vegeta and Trunks spent 2 days in the Chamber, which is consistent with the strange gap in time 174 presents.
Goku says they'll rest for 3 days, train in the other 3, and rest in the remaining 3. This didn't end up happening, as he was searching for Dragon Balls.
While there is some time (likely a few days), Broly taking place in an alternate timeline still makes more sense. Firstly, the narrator of Episode 174
heavily implies that the days were uneventful. Secondly, the movies are mostly considered an alternate timeline, and Broly never shows up as a villain in DBZ or GT, despite the inhabitants of Hell being resurrected in the latter series (Cooler, for example, shows up in GT). Lastly, it makes sense with the time scale presented in 174, where Vegeta comes out of the Time Chamber.
Given Toriyama's statement and the anime giving absolutely no support for Broly's existence, I think direct evidence that it's canon is needed, rather than claims that "it's not contradictory".