didn't u quote "Let T denote the amount that the government collects from households in taxes minus the amount it pays back to households in the form of transfer payments"?

so T= taxes on household - transferred payment = net taxes >0

T-G = public/governmental savings, which is governmental income - governmental expenditure,
Y-T-C = private savings, which is income - net taxes - consumption.

the author indicates nowhere that T=0, and the formulas totally dont need T=0 to hold valid.

卧槽，我傻逼了，谢谢大大！！

大大，这一章我还有一个问题，现在太晚了，改天我再请教你哈，谢谢！

大大，我第二个问题已经补充了，请你看看。

hmm, guess we are reading different versions of Mankiw's.
But I believe he made some assumptions which u may have missed.

First of all, in a close economy, Mankiw assumed that GDP or total output is fixed by factors. In another words, think it as Y is determined by physical capital P and labor L, and under this situation, he assumes that either P or L or both is being used or applied saturated, which means u've already been using 100% of P or L or both, so Y is totally fixed at the maximum and cannot change.
Secondly, he assumes the government doesn't increase taxes T, therefore, the household consumption C=C(Y-T), as a function of Y and T is also fixed, therefore, private saving PS=Y-C-T is unchanged too.

Now look at the equation:
Y-C-T+ T-G =I, if the government applies an expansionary fiscal policy, which means G increases and they need to issue more sovereign bonds to borrow, then investment I has to decrease.
If u think of the left side of the equation as the sum of private savings and public savings, or "the source of loanable funds", then since PS=Y-C-T is fixed, and T is fixed, the increase of G decreases the total amount on left side of equation, which is reducing the supply of loanable funds.

suppose now some households want to borrow some money. what would they borrow for? there are only 3 ways to put money into: consumption, saving and investment. But remember now we have the constraint that Y,T are fixed, hence C and PS are also fixed, therefore the money can only go to investment I, which means u increase the right side of the equation, that's increasing the demand. it's easy to see that u don't borrow to save in bank, but the everyday experience of borrowing to consume may be in your way here. just remember C is determined by Y and T.
also by households borrowing, Mankiw may actually mean business borrowing, because the concept of firm is actually totally omitted in this model, and ultimately firms are owned by households anyways, so u don't need to think about firms here.

if u change the equation to:
Y-C-T = I + G-T, then that's what u said that the government is borrowing from the households to fund its running, namely, to finance its expenditure or deficit, and this borrowing, given that private savings,Y-C-T,is fixed, would crowd out I, since now government uses bigger proportion of private savings.
pretty much these are just different interpretations of the same paradigm.