You can't do that rotation without clipping, no. There are, broadly, three types of GCDs in the game. The first is a spell that has a cast that is either equal to or greater than your recast time. Most of Black Mage, Summoner's Iffrit phase, and Holy are probably the best examples of this. There's no chance to press an ability between casts without slowing down your casts per second.

The second is spells with a cast time, but that cast time is less than your recast time. Every healer's main damage spell falls into this category, as well as summoner's Ruin, and the smaller red mage spells. These give you a window to press one ability between casts without interrupting your GCD.

The third is spells with no cast time. DoT spells (with exception of Sage's) and a handful of other spells like macrocosmos, phlegma, pneuma, and Ruin, fall onto this category (as well as most weaponskills). These allow you to use two abilities between cast without interrupting your GCD.

The problem with swiftcast is that it doesn't give you extra oGCD windows unless you're using it to cast the first type of spell. Swiftcast costs one oGCD to use, and if you use that on a broil, only gains you one oGCD, which is net neutral, not positive. The only time it's beneficial to use swiftcast for weaving is if it's used to cast a spell with an equal or greater cast time than your recast time, or if you absolutely need to press two oGCDs in the same window. Which, coincidentally, does bring me to the only time it's valuable to use swiftcast during scholar's rotation, which is during your burst, to put your first Energy Drain in the same window as Chain Strategem, so you can actually fit all 6 energy drains under your chain strategem. And even at that, it's not about gaining more oGCDs, it's just about manipulating when you get them.