Is that even legal to do? Can someone create their own centralised cryptocurrency and let people buy/use it?
What is legal/not-legal can change drastically depending on what jurisdiction you live within. If you have legal concerns, you should talk to a lawyer.
Generally though, there is a history of powerful governments shutting down centralized digital-money systems (eg Liberty Reserve), which culminated in the invention of the Bitcoin network (which has no central point of control/authority which can be shut down). This is similar to the evolution of file sharing: tools that were still at least partially centralized were shut down over time (eg Napster), until eventually someone invented a tool that couldn't be shut down (bit torrent).
Everyone who participates in the system agrees to trust this 1 party only, because they find them trustworthy
Even if the operator of the currency is perfectly benevolent and fair (ie, trustworthy), there is still a massive risk inherent to having a centralized point of control: what happens if the control falls into the wrong hands? (eg a rogue employee, hacker, government, etc). This is a massive risk, that only increases as the value secured by the network increases. What happens if the government policy changes? A war breaks out? The centralized entity running the network goes bankrupt? What if a natural disaster takes out their office? There are a lot of ways things could go wrong, and even if the chance of something happening is small, the impact could be massive.
On the other hand, the Bitcoin network has no hierarchy, no admin, no central point of control or authority. On the contrary, the network is governed from the bottom up, making it much more robust against many types of attacks.
Or they can just process transaction on a 1st come 1st serve basis and eliminate the need for blocks, just timestamp each transaction :)
This would basically amount to an .csv file with limited write access, which is rather uninteresting in light of the above.