Internal power supplies, while not universal, were not unusual at the time. The major considerations were cost, ease of design, and safety (that is, passing safety regulations that already existed in most first-world nations), with thermal considerations factoring in somewhere further down the list. The latter tended to point towards implementing a switch-mode PSU rather than a linear one; though a linear PSU tended to be cheaper, it was generally far less efficient. Some early computers, such as the Commodore PET, did use linear PSUs, due to the relative ease of designing them using off-the-shelf components.
When the computer and its monitor were sold as a set, it made plenty of sense to tap the computer's power off the transformer that the more power-hungry monitor needed anyway, and integrate only a rectifier and regulator into the computer proper. A variation of this principle also applies to all-in-one computers such as the Amstrad PCW, the original Macintosh and the PET. The computer logic, at the time, only needed a few watts at most, so this was quite a sensible option.
However, that was not an option for computers designed to attach to some random TV that the user already owned. The PC clone industry also demanded that monitors be interchangeable between computers, so a PSU had to be dedicated to the computer rather than shared with the monitor.
Safety certification required that the user should not have physical access to mains-voltage circuits, even when opening the case of the computer to tinker with the logic circuitry (as was then commonly expected). Putting the transformer in an external box, to replace the tap off a monitor's transformer, was one logical solution. The practice survives today with laptops and phone chargers, mostly to get the weight of the mains-voltage circuitry out of the computer itself.
The BBC Micro serves as a good example of how thermal considerations could be solved by using a switch-mode PSU. Early prototypes used a linear PSU, and there were concerns about how warm it got inside the case. Most of the logic didn't mind being a little above room temperature, though one of the custom ULAs proved to be decidedly marginal in the final design. However, design of a switch-mode PSU module, encased in a stout metal shield to deter user meddling, was subcontracted to a third party and proved to be both much more efficient and sufficiently inexpensive to meet the production budget for the machine. The production BBC Micro scarcely gets warm inside.
PSU modules of this type, designed by companies specialising in PSU design and safety certification, became standard practice and were commonly used in PCs and other microcomputers from the 1980s onwards.