There were 'some' (and by this I mean I don't have personal experience of this to speak of) concerns about 2 things with regard to synthetics. One was the belief that they were 'so' slippery that with roller lifters the roller wouldn't roll, it'd literally 'slide' or the lobe would slide across it, on roller cams (which many engines - even pushrod engines - were going to - i.e. roller lifters, back in the 'relatively' more early days of widely available commercial synthetics. I have no personal experience of this save for the fact that I know of plenty of people who have run roller lifter based engines (mostly gm gen ii/iii and beyond v8s) with synthetic oil without a problem. And for all intents and purposes you'd typically find the slipperier the better.

the other concern initially was that under extreme heat (the het of blowby, burning gases, and also the heat which was both partly blowby related and partly inherent design related) on air cooled engines. The fear was that some of the friction modifiers and VI improvers used to make the early synthetics even more 'multigrade' (staying the same thickness over a wider temperature range) - the fear was that these bits would 'break down' and then the oil you are left with is inferior to (fresh, low mileage) mineral or monograde oil,

Pretty much all of this second concern is no longer even potentially an issue because modern generation synthetics aren't 'full' of those particular additives any more, but rather have been refined (as in 'developed' not as in extracted from the regular mineral based oil refinery process) to the point they are stable across a wide range of temps..

auto trans oils are a good choice, esp if you are on a budget, they are the right sort of viscosity, and _will_ have an additive package to suit the gears and bearings in a supercharger, and also have some anti-frothing agents in there that will tend to help the oil get to and stay where it needs to be.

I tend to recommend the dexron spec trans fluids becuase they differ to the tqf (I think also called type f by some oil co's) spec oil a little bit in their friction attributes, but mainly because dexron tends to be fine with all types of oil seals and gaskets, but the other can cause deterioration of some particular type of seal (some particular chemical ingredient in the seal). For the record I don't know what that specifically is, but rather have heard about it leading to seepage/slight leaks if TQF was put into a trans requiring dexron, but not the other way around. The two fluid types do differ in how the clutch packs 'grab' when a particular band or clutch is engaging the next gear in an auto. I think (but wouldn't swear to it) that some people tried tqf in their auto (which was meant to run dexron) because it would grab and engage the next gear slightly more aggressively. which is possibly how people found this out I really don't know for certain on that front, but I can say that if i was ever going to use auto trans fluid in a manual gearbox or power steering or a supercharger like this, i'd run a dexron spec fluid for sure.