MMT "theorists" will say different things every time you ask them.
I doubt many of them would say this though, because it's pretty obviously on-face wrong.
But as you've stated it, the "theory" is on-face wrong.
Money isn't created when the government spends because it borrows from private credit markets when it spends, taking money out of circulation.
Money is created by monetary policy, not fiscal policy.
I don't necessarily disrespect MMT, I just have trouble tying people down to making a specific claim. Either they are saying something obvious and not at all contrary to mainstream economics, or they are making claims that I think are wrong - but I have trouble pinning down any particular claims.
MMT proponents would not character bond issuance as something that takes money out of circulation per se. It is only when that money is taxed back out of the economy that money is removed.
In this construction central bank policy and government budgets both do play a role, the question is what comes first. When governments runs deficits, it signals the central bank to print money.