You don't need to wait until you're down to 10 BBs before you start shoving and should actually be shoving much sooner before you get to 10 BBs.
The reason 10 BBs is a good, basic starting point for playing a shove/fold strategy is because:
-10 BBs still gives you plenty of fold equity
-You should never be limping in when you have 10 BBs
-Nor should you be raise/folding when you have 10 BBs
-So, if you're in a position where you're raise/calling, then you might as well be the first one to jam it in.
-Please not that 10 BBs is a great stack to open jam with, but not necessarily coming over the top of another players raise as they'll usually be getting good odds
to call depending on how much is in the pot already. You're never going to get anyone to fold if you shove 10 BBs after they put in 3-4.
There is almost no difference with jamming 12 BBs vs 10 BBs, so don't think you have to wait until you get down to 10 before making this play. Depending on the structure and how fast blinds increase, you should be jamming much sooner when you have hands worth jamming. The faster the blinds increase, the much sooner you should be jamming as your stack will get chipped down much faster while either waiting for a hand or waiting for you stack to drop down to 10 BBs. Sometimes, you're at a table where you have to wait and it sucks, but if you can help it, it's much better to jam much sooner with a stack over 10 BBs than it is to wait.
5n a turbo MTT with 5 min blinds, I might open jam 15-18+ BBs with 1010 in MP with high blinds and antes. Normal structure games that are 10-12 minute blinds sound like you have plenty of time to wait for hands, but you really don't. Especially when blinds and antes are high as you're paying almost 2.5 BBs per rotation. If you're sitting on 15 BBs, in 2 rotations, you're down to 10 BBs, less when blinds increase, and less BBs = less fold equity.
Fold equity is very important when in shove/fold situations. It's pretty easy to shove hands like KK, AA, AKs when you have 10 BBs, however, there are situations where we don't pick up these hands and have to shove much weaker ones like QJo or Q9s. If we have fold equity, it makes it harder for a player to call with a hand that might beat us preflop. For example, if we open jam 10 BBs with Q9s, it might be hard for a player to call with a hand like A4o, even though they have a slight edge over us. However, if we have no fold equity and we're only jamming 4-5 BBs with Q9o, a player can easily call with a hand like A4o.
So, to answer your question, 10 BBs is place to start playing a shove/fold, but don't be afraid to shove more. I'm willing jamming 25 BBs in certain situations. Above, I mentioned that you should be open jamming with 10 BBs, but 15-25 BBs is the perfect stack size to not only open raise, but to be re-raising all in over another players raise with as well. Especially in situations where there is a lot of dead money in the pot. For example:
-There are a lot of limpers and you're in the BB with a decent hand.
-There is an EP raise from a loose and 1-2 callers before you. You can squeeze play in this situation and pick up a nice pot.
-A shorter stack goes all in and you have a strong hand that beats their shove range. Jam it in to isolate with that player.