The VT100 terminal already had an 8080 processor.
So why did it have to be connected to a host computer? Why not use its processor to perform computations?
Is it because the VT100 processor's purpose was only to transmit input from the keyboard to the host computer, receive & display the corresponding output to the screen and to render the command line interface on the screen?
Was it also because the host computer was much more powerful? If yes, how powerful was it compared to an 8080?
"Having a processor" is not sufficient for something to be a general purpose computing device. The VT100 was designed for a specific purpose, and adding more features like storage and I/O would have made it more expensive for a purpose that most customers would not want or need.
At the same time (late 1970s), other people were indeed building general purpose CP/M machines with 8080 processors (eg. IMSAI 8080 was an early one, there are many others).
So, the answer to "Why didn't DEC do this?" is essentially "Because DEC's customers did not want that."
The 2MHz Intel 8080 ran at 290 thousand IPS (instructions per second), whereas the VAX 11/780 that a VT100 plugged into ran at 1,000 thousand IPS (aka 1 MIPS). Thus, 3+x faster.
Just as importantly, the VAX had much higher IO throughput than anything designed around the 8080.
And, of course, the VAX was 32-bit, while the 8080 was 8-bit. Even slow PDP-11 systems were faster, because of the better IO capacity.
So why did it have to be connected to a host computer? Why not use its processor to perform computations?
Because a VT100, or terminals in general, are dedicated devices for a specific purpose. Having a processor doesn't mean it's maent to be a computer.
The very same thing why one doesn't play DOOM (*1) on a washing machine, a fridge or your cars engine. All of them got at least one processor (*2) but are not computers.
The same way the purpose the VT100 Terminal was build for is to be a Terminal, not a computer.
In fact, I'd say that today, with most applications build around the web, most PCs are used almost exclusive as terminals - after all, a web browser is nothing else than a somewhat bloated ... err ... advanced terminal application.
When writing this question, you didn't use your PC (*3) as a computer, but as a terminal to connect to the massive computer network we run RC.SE on (*4).
Heck, that question has even potential for recursion:
Why does the PC need an 8088 when there is already an 8048 Keyboard controller?
Why does it need that 8048, when there is already an 8048 within the keyboard?_
Even more, there's another one in the mouse!
Bottom line: Having a processor is in next to all cases just a way to replace hardware (circuitry) by software, enabled due the universal nature of processors, but the purpose is usually not creating an universal device.
*1 - Yes, I know, it's a fun to make such devices play DOOM anyway :))
*2 - If not a whole lot more CPUs
*3 - Or whatever device you wrote this question with.
*4 - Well, kind of :))
Digital did make a standalone VT100, but it didn't use the built-in 8080. It used a 12-bit Intersil 6100 CPU chip and could run PDP-8 programs such as WPS-8. It was called the Digital DECmate VT278.
At the beginning of the '70s it was more the limitations of memory than the processing power that was what was a terminal or what was a computer. You can see it when you study what early glass teletypes are made of. Even the Apple I used shift registers instead of RAM for the glass teletype part of the circuit. The datapoint 2200, from which the 8008 was derived, was relatively compact even with its TTL CPU but its 2K of memory was of the serial shift register. The computer revolution of the 70s/80s has more to thank to the progress in memory technology than to CPU's imho.
What would you do with it? Its only connection was an RS-232 serial port, which connected it to a mainframe or minicomputer. It didn't have a disk for program storage, or even a cassette interface like a TRS-80, a popular personal computer of the time.
It was about the same size of a TRS-80, so they could potentially have turned it into a personal computer, but then it would be much more expensive. DEC made most of their money from sales of computers, which needed terminals. They'd still need terminals for those customers, and turning them into full-fledged microcomputers would not make them significantly more useful in that role.
This is hinted at in other answers, but it really is the key. There were two ways that a video terminal (or a Teletype, for that matter) could be used:
A terminal could be a computer. The Datapoint 2200 is one of the earliest examples. At its heart it was a terminal. But plenty of things could be added, including a hard disk drive, floppy disk drive, network card and additional RAM, so that it could indeed be used as a standalone computer. The 2200 used one processor, which was the basic design for the Intel 8008, to run everything.
A terminal could also include a computer in the same box. An example of that is the Heath H-89/Zenith Z-89 which used a Z-80-based video terminal as the housing for a full CP/M (and other OS) computer. For a non-technical user, it was one magic box. For a technical user it was a computer and a terminal combined together.
Many other computers, commonly running CP/M but not exclusively, had one box (or sometimes more than one) with a computer and a terminal, such as a VT-100, connected via a serial port.
In all of the above cases, this was one user = one computer.
This was the primary use case for VT-100 terminals and, I would argue (but don't have the stats to prove it) for most serial terminals. A single computer, from PDP to VAX to various microcomputers (8080 or Z-80 with MP/M, 8086 and above with MP/M-86, Concurrent DOS, PC/MOS and other operating systems, plus numerous other systems from Data General, Alpha Micro and others) could connect to anywhere from 2 to dozens of serial terminals. While a multi-user system could be used as a bunch of nominally separate user spaces, sharing the cost of CPU/hard disk/etc., the big advantage of a multi-user system was (and still is today, though more commonly in the form of a networked system) sharing files (instead of having to physically transmit or copy on a floppy disk, etc.) and sharing applications (e.g., accounting, inventory tracking, etc. all using the same data).
If you are DEC then you want to (a) sell more PDP and VAX systems with lots of terminals along with the systems and (b) sell lots of terminals to competing multi-user system vendors by building a high-quality, standard setting terminal such as the VT-100.
Many large multi-user system vendors sold either their own terminal or a rebadged version of someone else's terminal. WYSE eventually made some of their own computers, but their main product for many years was terminals that could be connected to other vendor's computers and emulate the native terminals, including DEC terminals.
The vt100 VT 100 specs had a nice display, a nice keyboard, a RS232 interface, a slow processor, 8 KB of program in ROM and 3KB of RAM. You could probably use it as a calculator without RAM and a storage unit.
Always remember that technical stuff is always a compromize between conflicting goals. In this case, price of additional resources and the concept that a terminal should be, exactly, a terminal connected to something bigger.
At the time, 1978-ish, a Megabyte of Ram costed about 10.000 USD Ram prices. A small harddisk probably about 500 dollars storage prices
So why did it have to be connected to a host computer? Why not use its processor to perform computations?
What wasn't mentioned here before was the other direction:
Many computers were JUST computers, that came with storage devices (sometimes not even that), but without any devices for human interaction. So the host computer needed a terminal to be connected, and often several terminals where connected to the same host.
