I think your subject title goes a bit over the top, this doesn't solve ALL problems.
Anyone is free to lockup collateral voluntarily, and all accounts have the option of a public balance.
I forgot to mention a couple more points. You would make the collateral bids not rollover automatically, that way you can't get stuck with zombie delegates you need to vote out as well. The locked collateral can't be used to vote for anyone else too. This would make the system more resilient to attack. To prevent collateral bid spam from flooding the system, there would need to be either a minimum amount required and/or or a small proof of burn probably.
This solves many problems with DPoS because:
1) You have to assume the majority of voters are lazy and voter apathy will be the majority not minority. There is no current guard against this.
2) There has to be some type of automated delegate selection method to deal with the voter apathy problem, which currently doesn't exist. You can still manually vote to avoid an aristocracy if you want, so the base voting method is still preserved.
3) Due to voter apathy, there is no way to easily get rid of zombie delegates, that fixes this as well.
4) Collateral is an easy to utilize sybil prevention mechanism. A flat rate only puts an upper limit on number of sybil nodes, while a bid system is more efficient.