Yes, but very limited. Basically, only if you're disabled, old (basically the retired), or temporarily residing elsewhere (aka rich people with multiple homes).

Who is entitled to vote absentee by mail?

1. Any person who is temporarily residing outside of their county of residence, and the ballot must be mailed to an address outside the municipality.
2. Any person who has a temporary or permanent physical disability and who, because of such disability, is unable to vote in person without substantial hardship to himself, herself or others, or whose attendance at the voting place could reasonably cause danger to himself, herself or others.
3. The parent, spouse or dependent of a person with a temporary or permanent physical disability who is hospitalized outside of his or her county of residence or more than fifty (50) miles distant from his or her residence, if the parent, spouse or dependent will be with such person on election day.
4. Any person who is sixty-five (65) years of age or older.
5. Any incarcerated individual who has not been convicted of a disenfranchising crime. Residency of incarcerated individuals is determined by the residence of the person prior to his/her incarceration.

https://www.sos.ms.gov/yall-vote/absentee-voting-information

Better ways to handle it though i.e. send ballots out x weeks beforehand and demand they are returned by polling day.

because each State sets how their elections are handled. Many states do this, but mail-in voting isn't universal. Some states as soon as you register they're automatically going to send you a ballot you can mail back or drop off, others you can register for permanent early ballots. I bet you can guess which States those tend to be, and which political party tends to govern them. There are states that don't allow it at all, and some only allow you to request an early ballot if you can prove you're going to be out of the city/county/state on election day. And that requires that you weren't purged from the voter rolls without being notified after the registration deadline (which can be months before election day).

places with mandatory voting (e. G. Oz) work just as hard and manage to do their duty.

That requires that voting is made easily and readily available. Something not true across much of the US since there is a party that historically does better when fewer people vote. So they do everything they can to prevent people from voting as much as possible (legally or not). Things like shortening hours the polls are open, or removing polling locations entirely, even making it so there's only 1 location for an entire county with millions of people. Or not allowing people to vote after the polls close, even if they are already in line. And purging voter rolls days before the election so people have no idea they're no longer registered to vote.

Not to mention that while employers are usually required to give you time to vote, the requirement is often only an hour or two, while the lines at the polls can be multiple hours long. And that doesn't take into account people having to work multiple part-time jobs where neither is required to give that time because there should be off hours to vote, but with the other job/jobs that time doesn't actually exist.

we have postal votes in Australia and the closing date for them to be received is 2 weeks after polls close.

And? Our bureaucracy is the way it is by design, to favor slavers.

Ok I do my job and mail it back early. But the postal system doesn't drive it over quickly enough and it doesn't get there in time. So my vote doesn't count? I did my job. So why should my vote not count? As long as I do my job and it's postmarked by a certain date, my vote should count.