I added an answer mentioning the Kiwi online travel agency. I received some feedback that it may not to be the best online travel agency in the world. What potential issues may arise from using the Kiwi online travel agency?
I think the main issue most commentators may have on this topic is that it is felt Kiwi have a strong tendency to sell self-transfers with limited information about the many associated risks and limitations, while advertising a “guarantee” that is quite limited in scope.
Beyond that, more generally there is little love for OTAs (Online Travel Agents), and it is usually recommended to book directly with airlines (and other suppliers such as hotels), to avoid having an intermediary which only complicates and delays things as soon as there is any issue (especially when they start playing ping pong).
A (genuine) travel agent has a commercial relationship with the airline it acts as an agent for; to have ticketing authority, it must agree to various terms and conditions.
By doing so, the agent (online or otherwise) may gain access to discounted fares not otherwise available to the public. This is just another tool in airline revenue management - it provides an alternative channel to sell some inventory, whilst protecting the yields in other channels. Or the agency may choose to discount further on the flight component, because they intend to bundle in other products such as hotels which makes the package as a whole profitable to them. The end result is that an agent may be able to offer you the customer a cheaper price than going direct to the airline, which is attractive to you, and hence makes being an agent a viable business.
To maintain this business relationship, an agent has to agree to adhere to practices that the airlines like, but passengers may not. The reason an airline can have a pricing policy that makes one ways more expensive than returns, or a flight A-B-C-B-A cheaper than B-C-B, is because regular exploitation of "skiplagging" or hidden city ticketing by an agent will see that agent losing their ticketing authority.
The potential downside, as a customer, is that the agency owns the booking until close to departure, so you must approach them for servicing, not the airline. This can make life harder in the event of disruption or cancellation, but that depends on the quality of the agent: a huge amount of business travel is handled through agencies without issues. But a particularly cheap OTA might come with a worse level of customer service, which is why you'll see warnings against using them.
With Kiwi, the situation is much worse - they tried to have their cake and eat it, by being an agent yet deliberately offering such itineraries. As a result, some airlines refuse to work with them - here's an example of American, BA, Finnair, and Iberia announcing to other agents that kiwi is not allowed to sell their flights.
So without proper ticketing authority, kiwi would now have to make a direct booking with the airline, in your name but with their contact details (else you'll have control of the booking). If the airlines detect this, they may cancel the booking entirely.
If your flight is cancelled, the airline might be offering its passengers rerouting options. However, Kiwi owns your booking and will not display these options to you, despite their T&Cs 3.1.1 stating they will be "choosing the important information necessary for your enjoyment of the ordered Third-Party Services, and providing you with this information via our communication channels".
Example: last December my flight from Manchester to Brussels with Ryanair was cancelled. Ryanair permitted passengers who booked with them directly to choose an alternative itinerary, eg. fly to Paris an hour later instead. However, all that Kiwi displayed to me were Manchester-Brussels itineraries with other airlines on the next day (that you needed to pay full price for). Kiwi's answer was that they "do not have the same offer that the carrier has".
Kiwi normally adds their service fee on top of the price you can get with the carrier, so there is rarely a point in using them, but sometimes that fee is negative (!), and that is most often the case with Ryanair. Also, these days, apparently, Ryanair doesn't permit Kiwi to do their automatic check-in for you anymore, and after you book a Ryanair flight through Kiwi you need to verify directly with the airline, thus becoming the owner of the booking. If your flight is cheaper through Kiwi and you're sure you won't need to add "services" such as luggage or seat booking, you can go through Kiwi. You will get your flight with some token discount which will be a loss to Kiwi (which they deserve).