Is it possible for an economy and government to function without regressive taxes?
I believe this could severely restrict the tax base, but I am not sure if there could still be enough diversity amongst the progressive taxes to sustain the government and get the economy on check. I also suppose this could be partially solved by raising rates.
Is it possible for an economy and government to function without regressive taxes?
Trivially it is possible. Government could in principle just collect some revenue through progressive income tax.
I believe this could severely restrict the tax base,
No, typically the tax base is define as value of sum total of economic activity (e.g. income, wealth and so on) on which taxes can be or are levied. For example, for linear income tax $tY$ tax base is $Y$ not $tY$
Abolishing some taxes lowers government revenue but does not erode the tax base itself.
I am not sure if there could still be enough diversity amongst the progressive taxes to sustain the government and get the economy on check
It is unclear what you mean by government keeping economy on check. Typical government does not require a lot of revenue to sustain itself. For example, US spends only very small part of its budget on administration. Most of the budget is devoted to some sort of transfers that are not necessary for government to sustain itself. Even if we would consider all public goods essential you will still end up with most of the budget being transfers.
