To be absolutely clear... The only reason that the Democrats would 'pack the court' in this manner would be because the Republicans are currently trying to 'pack the court' with a pronounced conservative bent (explicitly, by preventing Obama from appointing a Justice when an opening became available in the last year of his second term, then pushing through a Justice in the last months of Trump's term). All the exaggerated political keening and pining aside, this is (frankly) straight-forward politics. The GOP is trying to use politicalization of the court as a means to ensure their political dominance into the future, for reasons of their own. If the Democrats take full control of the government in 2020, they will work to undo that, for reasons of their own. They could, and likely will, add as many justices and judges as they need to get slightly better than par.
As the wise men say: don't put all your eggs in one basket; don't count your chickens before they hatch; be careful about letting the fox into the henhouse. Do not ask me why so many of these aphorisms are chicken-related...
Congress can expand the Supreme Court at will: that is merely a matter of passing a bill, and if the Democrats take control of both Houses and the Presidency that is a non-issue. Congress can impeach and remove Supreme Court Justices at will: again, a non-issue if Democrats control both Houses (though one subject to the need to acquire a supermajority in the Senate). Congress is (again, frankly) in the driver's seat in US politics if it so chooses to be, but Republicans (as the minority party) have over the last 20 years or so have instituted a policy of obstructionism, shifting more and more power off to the President (when the President is Republican), while refusing to pass actual laws. They favor executive orders, because executive orders are easier to institute when favorable, and easier to obstruct when disliked, than any actual law. That works in the favor of the minority party, which Republicans have been for some time.
Since the current (pre-appointment) partisan split is either 5-3 o 4-4 (depending on how one counts Roberts), I expect the Democrats would aim for 11 or 13 Justices. That should restore balance without seeming overtly partisan. The goal is to appear as though one is addressing an injustice rather than packing the court; but that too is a matter of politics.
I haven't looked deeply into Barrett. Clearly the GOP hope is that Barrett will act as a conservative stooge, simply reiterating conservative ideology in judicial decisions. That is possible, but problematic. Any person nominated for the Supreme Court is going to be both intellectually sophisticated and painstakingly independent; they have they own reputation to think about, and their decisions will be less biased than partisans anticipate. But ultimately SCOTUS cannot match the power of Congress. Congress makes laws, and if the Supreme Court overturns this or that with an unfortunate ruling, Congress can write law to replace whatever the Supreme Court undoes. Packing the Court (by Democrats or Republicans) is in the end a losing scenario.