1: show up at polling location, either on voting day or within the early voting period.
2: show ID to poll worker who verifies it and hands over the ballots for the elections in which you have the right to vote
3: fill in the ballots, put them in the envelope(s) and hand these over to a poll worker
4: poll worker checks again whether you have voting rights for the specific elections and drops the envelopes into the correct voting urns
Done. No postal service involved. You can vote at any (early) voting location for national elections, for local ones you'll need to go to a local voting location where the required ballots are available. Early voting can start a few weeks before election day so nearly everyone should have the possibility to set aside half an hour to an hour to do their civic duty.
That leaves those who really can't make it to the polls. For these people there can be a verified postal ballot system, i.e. they get one sent home on request, fill it out and send it back in the required envelopes. Voting by mail always adds risks but since it would only be needed for a small fraction of the voters that risk is manageable. Mail-in votes need to be counted on election day so they need to be in the mail about a week before election day. Why you ask? Because the fraud risk of mail-in ballots makes it all too easy to use them to add enough votes to push a close race towards a specific candidate. Count them early on election day and publish the results, these need to be final results, before the rest of the votes are counted. There's plenty of countries where votes are counted this way, this is not rocket science. I have voting rights in 2 different countries in northern and western Europe, both use something resembling this scheme. Everyone shows his or her ID either to get the ballots or to have them deposited in the voting urns, this has never been contested and the mere thought of voting without ID is so clearly outrageously silly that nobody brings it up. All the arguments against showing ID have been proven to be specious at best, you need ID to do just about anything related to the government so insisting on voting without it only serves to undermine trust in elections.