No, I think this is more about a country's ability and willingness to control borders and impose quarantine restrictions.
Border control working to keep the virus under control
The trick to keeping the virus under control in at least China, Australia, Taiwan, New Zealand and Thailand is to get local transmission under control (if there was any) and then to make sure that no new cases come in and spread locally.
Thailand and China are countries with land borders, but they are still able to keep a hold of imported cases though extensive quarantine of foreign arrivals. This is confirmed by analyses from from foreign governments (e.g. the European Council) which classify these as low risk countries.
The same principle applies to Taiwan, New Zealand and Australia, but they have an easier job keeping control of imported cases because they are islands. That means most arrivals enter through a few ports of entry (airports, regular ports) where arrivals are also subject to quarantine and testing.
When it doesn't work
Of course being an island alone doesn't mean it's a safe haven automatically. The UK has had many coronavirus cases with significant local transmission. Once local transmission cannot be controlled, stringent entry controls won't prevent the virus from spreading.
And when you don't even have the benefit of few ports of entry, as many continental EU countries don't, then it's even harder to keep the virus under control. This essentially happened in the European Union with the first epicenter in Italy. Due to freedom of movement, many of those cases have spread to other European countries which failed to implement efficient quarantine for all travelers.
Going back in time, the Dutch state broadcaster NOS wrote on the 3rd of March 2020:
Het aantal bevestigde gevallen van het coronavirus in Nederland is gisteren toegenomen tot 23, meldde het RIVM. Dat zijn er vijf meer dan gisteren. Bijna alle mensen bij wie het coronavirus is vastgesteld, zijn reizigers afkomstig uit Noord-Italië of mensen uit een gezin van een eerdere patiënt.
Vanwege de uitbraak van het virus heeft het ministerie van Buitenlandse Zaken per direct het reisadvies voor Noord-Italië aangepast. Nederlanders moeten daar alleen nog naartoe reizen als dat noodzakelijk is, dus niet voor vakanties. Tot nu toe had het grootste deel van Noord-Italië de kleurcode geel. Dat wordt nu oranje.
The number of confirmed cases of the coronavirus in the Netherlands rose to 23 yesterday, the RIVM [Dutch health authority] reported. That's five more than yesterday. Almost all confirmed cases are travelers from Northern Italy or family members of previously confirmed patients.
Due to the outbreak of the virus, the Dutch Ministry of Foreign Affairs has immediately changed the travel advice for Northern Italy. Dutch nationals should only go there if travel is necessary, which does not include vacationing. Up to now, the largest part of Northern Italy was coded yellow [advice: beware of risks]. This will now change to code orange [advice: necessary trips only].
Back to Thailand, how did they manage it?
Comparing that with Thailand which has kept most local transmission under control until now (March 2021), there was already talk of quarantine for high-risk travelers as early as the 5th of March 2020:
On Tuesday, Thailand’s Public Health Minister Anutin Charnvirakul said in Facebook post that all visitors from “high-risk zones” would be required to quarantine themselves for 14 days, according to a Bangkok Post report. The post, which was quickly deleted “without explanation,” reportedly mentioned Singapore, Japan, China and South Korea.
Local lockdowns started April, with CNN reporting about the Thai tourist island of Phuket:
Restrictions are set to get even tighter. On April 9, the government announced a lockdown for all 17 sub-districts of the island for 14 days, asking all residents to stay home from April 13-26, "or until the situation improves."
"The move makes Phuket the first province in Thailand to impose a total lockdown of all areas, in its sustained effort to stop the spread of the Coronavirus Disease 2019 (Covid-19)," says a statement from the Tourism Authority of Thailand.
But entry from abroad was restricted from March 26th through an emergency decree requiring the quarantine measures that are still in effect today.
Back to the question
Is this necessarily an autocratic vs democratic situation? No, clearly democratic countries can take emergency measures too. And as we've seen (I've not shown a timeline from Australia or New Zealand because I'm not as familiar with their handeling specifically) some democracies have taken swift emergency measures whereas others (e.g. in the European Union) have reacted too slowly for such measures to work.
And while I've not given specific examples, failure to take measures quickly has also occurred in some autocratic countries.