You are describing what lead to the development of the Apple homecomputer: Steve Wozniak wanted to make a cheap video terminal, and ended up with a complete computer (except with the 6502, not the 8080). So no, it doesn't have to be connected to a host computer, but the itch that the VT100 wanted to scratch is "I want a video terminal instead of a teletype".

The IBM PC keyboard also had a processor in it and needed to be connected to the IBM PC to do useful work. The processor just scanned the key matrix and sent the button events to PC. Just because something has a processor to function does not mean it is a general computing device.

The computing power and memory available in these 1970s/early 1980s expensive* terminals was tiny compared to the mainframe/mini they connect to, so they can't run worthwhile software in the terminals. After that, what you're asking about is the advent of the microcomputer and the history of that is already very well documented and freely available on the internet. (*By modern computing standards and also at their contemporary costs for general businesses.)

also, "why does my smartphone need to connect to the internet if it already has a processor?" ;-)

@szulat, I think the smartphone example is counter-productive here and makes it far more confusing to readers. Clearly a smartphone with software installed is a high-powered and completely self-sufficient computer platform, with the internet an optional extension that expands its already ample capabilities. Put it in aeroplane mode, you can run happily compute away running practical application software on it all day. That's a thousand miles from an 8080 with a few kilobytes of RAM.

Apparently, DEC also thought of this, and some years later, when memory and other components were cheaper, created the DEC Rainbow, which was both a terminal, and a computer.

There's a processor in my router, but I doin't use it as a computer.

@Barmar Also: Printer, dishwasher, lighting dimmer, etc...

(Not sure enough to make this an answer…) In many cases, because it didn't have the programs and data you needed. The host computer is where all the data lived; it had access to tapes and disks and cards and printers and all the other storage and I/O devices; it could read all the data that operators had input, and accounts/ledger/etc. info. The host computer also had all the programs (batch or interactive) for updating and accessing that data, and for printing reports from it. (The exact details depend on the type of business, of course, but I think the principles applied quite broadly.)

@gidds Indeed, I was first going to say microwave, but I thought router would be a better example.

@Barmar OTOH, with the miniPC market of today, setting one up as a router is getting attractive. Not hard to find some with dual ethernet for this very purpose.

even your keyboard has its own CPU, that doesn't mean it's a full computer

@JoL There's a difference between using a computer as a router and using a router as a computer. People have been doing the former for many years. In fact, some of the first routers were minicomputers that were fitted with multiple NICs, before dedicated routers were produced.

@Barmar There is also a processor in many printers. Since Postscript is turing-complete, you can have fun in printing carefully crafted files.

@doneal24 Yeah, but it's still not a very useful computer, since it has limited I/O capabilities. You can't write an interactive application with it.

@Barmar Limited I/O but, in certain situations, able to initiate out-going network connections. There have been more than a few CVE's issued against PS in general and certain printers specifically. And with a active network connection I'm not positive that you couldn't have some degree of interactivity.

@ninjalj - which DEC then crippled by making their floppy format slightly different from the IBM PC so as to make it harder to move stuff around. Certainly kept my lab from buying one to try out since it would not play nice with our other computers.

Funnily enough, DEC did make an add-on board to convert the VT-100 into a general purpose CP/M machine. The upgraded VT-100 was informally known as the VT-180 or Robin. en.wikipedia.org/wiki/VT180

@Barmar there’s more to computing than interactivity.

@RonJohn That's true. I'm old enough to remember batch computing, which is certainly feasible with a Postscript printer.

@Barmar there's more than "interactive" and "batch". Daemons, for example, and single use systems like NICs.

Can you first explain your idea of a 'processor'? Might there be a difference between 'a processor…' and 'a processor powerful enough to do anything significant at the terminal level, rather than by connecting to the core?