The low number of travel exemptions for people who have recovered from COVID stems from the discussions around "immunity passports", which was one proposed solution to enabling travel prior to widespread availability of vaccines. The first counter-argument was that its unknown if infection can actually protect against subsequent reinfections:

There is currently no evidence that people who have recovered from COVID-19 and have antibodies are protected from a second infection.

This is now known to be categorically false, with natural infections being almost as good as a vaccine but its sort of ignored by most governments as they're trying to maximize the number of people getting a shot.

The other concern was that people would purposefully try to get infected so that they could unlock travel and other privileges:

Some experts say online immunity certificates are unethical as they would give certain privileges to people who’ve contracted Covid-19 and may even encourage people to catch the disease in order to obtain immune status.

At this time governments will at most grant some travel privileges to people who have tested positive for COVID in the past 6 months, i.e. Germany has the following policy:

How do you prove you’re recovered from Covid-19?

People who have recovered from coronavirus and were infected no more than six months ago also face less restrictions. The test must have been a PCR test (or similar), checked in a lab and taken at least 28 days ago, and be no older than six months. An antibody test is not sufficient because antibodies can decrease over time.

The last paragraph makes absolutely no sense as the presence of false negatives doesn't mean positive results couldn't be used to determine who's immune to COVID. Most likely this rule was put into place to justify the 6 months limit. With antibody tests people could potentially get around the need to get vaccinated for as long as their antibodies would last. Without them, they can only do this for 6 months after getting infected, which excludes everyone who got infected in the first COVID wave.

It can be frustrating when government policies don't follow logic and reason but unfortunately there's nothing we can do about it. Some countries go even farther - i.e. Canada doesn't give vaccinated people any travel benefits at all, not even to their own citizens with a Canadian vaccination certificate - let alone to people who have recovered from the disease.

NB: I am personally fully vaccinated and got my vaccine on the very first day of availability to my age group. This post is in no way intended to disparage the vaccination campaign.

An important point I think worth mentioning (and is related to the idea of decreasing antibodies over time mentioned in the other answers) is that the goal of vaccines isn't necessarily just to create antibodies; it's to trigger long-lasting (hopefully) changes in the immune system (e.g., the formation of Memory B cells). These changes will then allow the body to respond more rapidly to future infections by doing things like ramping up antibody production faster than it would otherwise. So even if antibodies fade over time, we still have protection for some time after that because of these other underlying changes. It's why some vaccines can produce life-long protection (for things like measles and chickenpox).

The problem from a testing point of view (for things like travel) is that antibodies are relatively easy to test for compared to these other things (with the caveats and issues mentioned in other answers applying). Some of the changes may be detectable in the blood, but to really know a persons true immunity would require digging into other parts of our overall immune system, like the lymphatic system.

Some reading topics for those interested in more details:

• Adaptive immune system
• Immunological memory

Something else to keep in mind is practical considerations related to antibody tests. Note that I'm not a doctor and as such might be wrong about certain things.

Antibody tests generally require bloodwork to be taken. Bloodwork means drawing blood from the patient. This brings many considerations compared to vaccinations and PCR tests related to how easily the test can be administered, who can administer it, how long it takes to administer it, where it can be administered and who is eligible to undergo these tests. There are also practical considerations related to the transport of the specimens and the test itself.

I'm not going to pretend like I know enough about these tests to know what these considerations are, but from the best of my understanding, people generally are more willing and able to get their sinuses/throat swabbed than they are to get blood drawn, and PCR tests generally are easier to scale up than bloodwork is. This may be part of the reason why countries may prefer PCR tests and vaccination reports over antibody tests.

Logistics & Economy

Medical tests require:

• Qualified crews.
• Supplies
• Time

Checking a document require a few seconds time of an entry level clerk.

Medical tests are way more expensive and disruptive than checking documents, and they escalate poorly.

I happen to be closely involved in the procedures for handling passengers at a local airport. Although they were advised to come with a PCR test results, those who did not have one would be subjected to an antibodies test.

And even with only a fraction of the passage affected, and with travel severely curtailed, it has already been a difficult task. You have to receive all the passengers, sort out who needs the tests, administer the tests, wait for the results, communicate them to the waiting passengers (and beware of privacy issues), getting sure that no passengers leave the testing area before getting the results...

If you want to allow travel on a significant basis, you just cannot rely on performing medical tests on arrival. It simply does not work. So you settle for the best next: assume that vaccinated people won't get infected, and let them pass freely.

Yes we do know that some vaccinated people may be infected and contagious, but even the risk of those who can be reinfected depends of the infection rate at their origin. So with low rates the governments are willing to bet on easing the restrictions on travel in order to estimulate the economy and appease the public. There is certainly an element of risk (as there has been any time restrictions have been eased) but they seem to expect to contain the situation at least until vaccination is high enough.

It is not that different from the current situation: even PCRs leave some room for not detecting infected people, yet may EU countries allow travel without quarantine with just a PCR. There is a need to balance the costs and risks of each safety measure.

When you acquire an infection, antibodies will show up. You are infectious at this stage.

Remember the children chicken pox parties of yesterday-year that were attempts at conveying natural herd immunity? That debacle lead to an increase of Shingles (an adult version of chicken pox) which can be very debilitating and even deadly. Virus can remain dormant.

Also, the Covid19 vaccines seem to convey some level of anti-spreading properties. That is, you are less likely to be a carrier and to spread the actual Covid19 infection. That is the rationale behind the CDC saying that fully vaccinated persons can go around maskless around other vaccinated persons.

As others have pointed out, the antibody tests are notoriously unreliable. I recall in the real news that a lady in the US was tested negative 2 times using the antibody tests but later was hospitalized for Covid19. During the the second and third tests, she was even exhibiting sever symptoms. The final 3rd test, a PCR test showed positive for Covid19.

With the super spreader UK117 variant floating around, the current vaccines may not convey sufficient immunity. I would continue to wear a mask when I am around a crowd. Virus do not recognize sovereignty.