I’d second the 1.005-1.010 range to target for all the above reasons. I wouldn’t worry about trying to be too exact, a 1.002 vs a 1.008 won’t make any noticeable difference IMO.
Thought I’d expand a little, however, on why not to wait too long to transfer.
Oxygen is good/necessary for the yeast while it is actively fermenting which is one of the main reasons for stirring regularly in the primary to introduce O2 (the other main reason is to ensure floating fruit does not have enough time above the must to grow mold). While the O2 is being used by the yeast, it won’t negatively impact the must.
Certainly, if you left the must in the primary well after fermentation ends, the CO2 will eventaually slow, stop, then dissipate and the wine is exposed with no protection. But in reality, you’re going to move to the secondary which is were O2 exposure is more likely to occur if you wait too long.
If you wait until your SG is 1.000, most of the yeast is done so the ferment is already weak. You’ll likely notice significantly less enthusiastic bubbling (in other words, far less CO2 production).
The simple act of filtering out the solids and transferring into the secondary will introduce a lot of O2 (unless you are super careful and doing so under introduced inert gas such as nitrogen or CO2).
Additionally, racking can temporarily stunt the ferment under normal conditions. If the must is already in a weakened state, it might be some time before the air gap in the secondary fills with CO2.
You want your wine to be active enough to make the journey with enough horsepower to keep the ferment going so your secondary airlock is fairly quickly bubbling every couple seconds at least.
One other item I’ll point out (unrelated) is that it sounds like you‘re sanitizing every time you open it to stir. This isn’t really necessary. Kudos for being anal on the cleanliness but you’re just adding work to what should be just 30 seconds of open-stir-sniff-admire-close. Oh, and don’t bother with the airlock in the primary. You need the lid to keep the fruit flies and other contaminants out (or towel/cheesecloth + rubber band) but, personally, I just use a loose lid, flip and place upside down, stir, and replace. Just my two cents.