The multiplier effect is a well-known economic fact taught at A-Level and if you don't understand it you really shouldn't be arguing economics.
It's simple enough to see in practice. If the government spends £100m to build a hospital, a lot of that will go in wages to the people who build it and in turn they'll spend some of that. There will be some 'leakage' as they pay tax (which increases government revenue) or buy imported goods but most will probably be spent locally. And the shops, pubs or restaurants they spend it in will also pay their staff. They might save some meaning banks will be able to lend a multiple of whatever they save. And so the cycle goes on. It's the same money being recycled but that process creates wealth.
But if the government doesn't spend that £100m then they end up paying welfare benefits, which mean much lower consumption and no revenue back in the form of taxes. So it entirely depends what you spend that borrowed money on and whether it's capital investment or paying out benefits.
You are misunderstanding though that your spending plans come from taxes and increased taxes are put there by growth. Currently we take say £1 in tax but we need £1.20, so we borrow the difference, 20p. If you increase spending and do not link that spending to growth then all you are doing is creating a bigger deficit. Yes you are increasing the tax intake but you are doing so not on the basis of growth but on the basis of a false economy funded by borrowing, welcome to Greece.
I won't even go into the fact that for every 1p you borrow you will repay more than that. Even today a big slice of your tax paid goes to absolutely nothing but interest. The current debt repayments are around £48bn a year so just imagine if we halved the national debt and suddenly had £24bn a year to spend... In fact reducing the national debt gives you extra public spending in that sense literally for free.
As I said, look at France, France is biggest public spender in Europe yet its economy is almost completely flat. In this situation would you say France should spend more in its public sector? What about Spain, Italy, Portugal, Greece who are all flatlining under massive budgetary debts. Should they all increase public spending to help reduce that debt...... Clearly not.
People often compare Germany to us and the reason we are in a pickle and Germany aren't is because Germany imposed immediate hard austerity upon 2008 hitting which enabled them to keep public debt under control.
They now responsibly spend their way out of the problem by linking that spending to growth. Once borrowing is under control they can do as you say which is to have spending bursts to invest in infrastruture, services etc to supplement that growth however if growth falls, spending falls, it is very simple.
Unfortunately if people want more public spending then you need to get your wallet out and start campaigning for increased taxes.... I can't see that getting far!