There are several "real" problems and several fake ones
Dead voters not getting purged from the rolls is usually due to poor processes, not corruption. Republicans looking to point to corruption want to review all the registration lists so they can point to all the dead people and say "see, here's all the fraud that helped Hillary win the popular vote" but in most places the voter registration rolls are not plugged into the social security "death master list". Many precincts all over the country have no way of knowing who died, so they simply remove voters from the rolls if they haven't voted for a while.
Besides, dead voters on the rolls are only a problem if they vote so you have to demonstrate not only that dead voters are on the rolls but they are marked as having voted after they died (several months after they died, as they could have sent in an absentee ballot) If you had CCTV cameras you could possibly identify them as they walk to the desk with the nice old ladies who check you off the list when you vote, but if you vote absentee for a dead person it would be more difficult to catch you. Though obviously coming up with a better way to remove people when they die the best solution - you just have to make sure it isn't doing more than that.
Which brings up what is more likely the biggest source of potential fraud - absentee ballots. To start with, they're less secure since there's no 'in-person' element. Just a signature, and signatures are completely 100% worthless for proving you are who you say you are. It is all too easy to send in a request for an absentee ballot for someone else, so long as you have a way to intercept their mail. That's simple for family members / spouses, or people who live in apartments where the locks on the communal mailboxes are trivial to pick. Not only that, but since most states don't share information on their vote registration rolls and who has voted, people who maintain residences in multiple states can easily vote twice with little chance of detection. Or even if they only vote once, they can vote in the state where they think their vote will matter more - one has to wonder how many of the votes cast in Florida in presidential elections are people who legally should be voting in their home state.
Obviously hacking (whether by Russians, political parties, or griefers) is a concern, as is trust in the companies making the machines, but that part should be the easiest to address. Require all votes leave a paper trail, and using that paper trail require a recount of a 2% of every state's precincts selected randomly (plus maybe allow each political party to choose a few precincts if they feel there are some that need checking due to election day irregularities) and if the tallies differ by some small amount require a full statewide recount. Heck, a full hand recount wouldn't be that hard after EVERY election, if you only recounted the president, senator and congressman, and not all the rest. Whether to recount those would be governed by state or local laws anyway.