Even then, if they hold the state delegations, that guarantees them control of the Executive for another 4 years. If the Dems picked their VP to be Biden, Warren, Sanders, etc --- they're just not going to be able to do anything but go settle some ties in the senate (their only real constitutional mandate). Anyone that isn't Pence / a Republican is just going to be relegated to join the other (MANY) useless VP's who just sit around and wait for their President to die.

The Atlantic has a good piece on this - https://www.theatlantic.com/ideas/archive/2019/07/what-happens-if-2020-election-tie/593608/ - Talks about how close some of the delegations are (on both sides) and how the delegation race could be decided by one congressional district in many states.

Presumably, the Senate would know in advance who the House will elect as President, so if Biden is going to be elected by the House, the Senate would probably elect Trump as VP.

Given Biden's mental state, he is likely to be much more than a figurehead President, and will probably not be able to serve his entire term as just a figurehead.

If Biden is President and Trump is VP, you may actually see democrats calling for Biden to either be impeached or to step down so Trump would not be eligible for a 3rd term if he serves less than 2 years as actual President after the 2020 election.

If you think politics were filled with drama today....