With Leonard they needed a lucky shot to even make it to the conference finals, where they already got before.
Well both the Bucks and 76ers were far better teams than any the Raptors had beaten in their previous runs.
With regards to your other points, this season's B3 finalists both had 7-8 good players on their roster (and it was arguably South Dragon's 6th man who won him the title). They didn't use their bench that much, because like you said, without LCD/blank, there really isn't much advantage to depth. However, if the situation was reverted to how it was 20 seasons ago, I'd imagine these teams would still be very successful.
Also, I looked at how long South Dragons was playing his starters per week and it generally seemed to hover around 65-70 minutes/week during the regular season. During playoffs of course it was higher (around 80), but this happens even in the NBA, where lineups shorten in the playoffs and stars take on a higher load (often averaging 35-40+ minutes per game).