Are there disadvantages to longer rests between sets?
Sure. I cool down if I wait longer than five or so minutes between heavy squat sets, and that can be a problem if my mobility is iffy and I really need to be warm to get good form. It's also annoying to have the two-hour-plus workouts that result from 10-minute rests between sets of, say, heavy squats and ego-driven pull-up sets.
Lets say I only barely make it through my first set of squats, and am sure that I will fail on the second set if I go for it in another 3-4 minutes. Is there any particular disadvantage other than convenience to, say, taking a break and doing my press, and then coming back with my legs completely refreshed for the next set of squats. Basically, would it be better to just to go with the shorter rest times and do 5/4/2 rather than wait as long as I need to and get 5/5/5?
All the problems above apply. Plus, I know you said this is purely for strength, but...there's a whole host of other benefits from being able to do the sets with only a five minute rest: improved muscular endurance, cardio, recovery capacity, mental fortitude to fight through soreness, and so on. Plus I find it a little more challenging to stay focused on my squat form when I'm switching back and forth with other exercises.
If you're doing a novice linear progression, I'd just focus on doing better with the 3 sets in a row, with 5 to 10 minutes rest, rather than trying to continue to add weight by jumbling the exercises you're doing.
This seems especially relevant with chinups/pullups, where I can either go to failure and do ~7/4/3, but if I wait a half hour between sets and go for sets of 5, I can easily total 40-50 chinups in a single day.
Now we're talking about Greasing the Groove and not a single workout. That's a whole different enchilada that requires separate research. It works, but single high-rep sets are more impressive and productive, as discussed in detail here.
For your pull-ups, I'd do both: three maximum-rep sets (with good form) with a maximum of five minutes rest, and sets of 5 throughout the day on rest days.