Are U.S. citizens barred from traveling to Indonesia?

Covid is not "over". Over the past 7 days there have been over half a million new cases of COVID-19 in the USA, and over 10,000 deaths. In Indonesia it's over 10,000 (reported) new cases, and over 500 deaths.

Each country is able to set it's own rules around entry of foreign citizens. Indonesia is currently only allowing select foreign citizens to enter - which generally does not include US citizens.

Timatic (the system airlines use to determine who is allowed travel) states the following for US Citizens travelling to Indonesia at this time :

Admission and Transit Restrictions: Passengers are not allowed to enter.

Information: This does not apply to passengers with a diplomatic or service visa. Information: This does not apply to passengers with an e-visit visa with a visa number ending with LN. Information: This does not apply to passengers with an APEC Business Travel Card. Information: This does not apply to passengers traveling under the Travel Corridor Arrangement (TCA).

Unless you meet one of the exeptions above, then at this point you will not be allowed fly. You will be denied boarding by the airline at your point of origin.

Passengers are always responsible for determining visa and entry requirements for a country when purchasing a ticket. The airline will sell you the ticket as they will presume you have done your own research and determined that you are allowed fly (eg, you meet one of the exceptions, or perhaps you are also an citizen of another country where the rules may be different).

Of course, this is a rapidly changing situations. Depending on when you are planning to travel it's possible that the rules will have changed by the time you are due to depart, in which case the rules in place at that time are obviously what will apply.

According to Timatic, these are the admission restrictions:

Admission and Transit Restrictions:

Passengers are not allowed to enter. Information: This does not apply to passengers with a diplomatic or service visa. Information: This does not apply to passengers with an e-visit visa with a visa number ending with LN. Information: This does not apply to passengers with an APEC Business Travel Card. Information: This does not apply to passengers traveling under the Travel Corridor Arrangement (TCA).

These are the Covid measures stated in Timatic:

Passengers must have a printed negative COVID-19 PCR or RT-PCR test taken at most 72 hours before departure from the first embarkation point. The test result must be in English.

Passengers and airline crew must have a COVID-19 vaccination certificate showing that they were fully vaccinated at least 14 days before departure. The certificate must be issued in both English and a local language of the country where the passenger was vaccinated. Exception: This does not apply to passengers younger than 18 years. Information: This does not apply to passengers with a diplomatic or service visa and an invitation letter issued by an Indonesian Ministry traveling under the Travel Corridor Arrangement (TCA).

Passengers are subject to COVID-19 PCR test upon arrival and quarantine for 5 days. Information: This does not apply to passengers with a diplomatic or service visa and an invitation letter issued by an Indonesian Ministry traveling under the Travel Corridor Arrangement (TCA).

A completed "electronic health awareness card (e-HAC)" must be presented upon arrival. The card can be obtained before departure through the "Pedulilindungi" mobile app.

Furthermore, it says a visa is required.

To answer your question, no you aren't allowed to travel to Indonesia unless you meet one of the exemptions in the first quote above.

Also, I called American Airlines, and was able to book a ticket to Jakarta. So, I got the ticket, what’s going to happen when I reach Jakarta? Are they going to send me home ?

If you don't meet the criteria to enter Indonesia then American Airlines shouldn't allow you to board. When you check in, the airline checks that you meet the criteria for entering the country. If you don't meet the criteria, the airline should refuse you. If they somehow let you board and you get refused in Indonesia then they might send you home.

I would suggest that you explore the option of getting a visa and abiding by the Covid regulations to allow travel as planned. Alternatively, you might be able to postpone your travel depending on the ticket conditions or any leniency the airline offers in light of the pandemic.