Recommendation
Since you want a bean-to-cup brew machine, you're limiting yourself quite a bit in terms of choice. That can be a good thing as it narrows down your options.
When to bean-to-cup brew machines I'd recommend the Breville Grind Control™. It's not a commercial-size machine, but with only 6 people that would probably make it needlessly expensive.
Since I have no first hand experience with the machine, I recommend watching this YouTube video, again by SCG, to get a good impression of the machine. Should you have machine-specific questions then I recommend asking those as a separate question.
As for price, it's advertised at around 300$ on Breville's website.
As for capacity, it's 60 oz (1.77L) machine. That means you can get 7 decent cups (250 mL / 8.45 oz) out of it.
You have some control on strength (in practice, more or less coffee going in) and grind size (finer or coarser grind). Furthermore, the can is insulated so your coffee keeps warm longer as opposed to machines that keep the can at a certain temperature (which eventually impacts taste).
The only criterium it doesn't meet is the ability to connect it to your water line. You might be able to make it work somehow, but it's not a feature of this machine (as opposed to others which are designed to be built into a kitchen or something).
For non-Americans: note that the machine is rated for 110-120 volts. It may not work in places with higher voltages and I'm not sure if there's a higher voltage version of this machine. Just something to consider before buying.
My answer consists of three parts. The first is an on-topic advice. The second is an option to get more information to better inform you which could be used to edit / ask a new question that is more refined. The last part is more of a comment which you should take note of in a purchasing situation.
Advice
Without knowing any more specifics, I understand that you don't want to spend a lot of money and that brew machines worked in the past but lacked capacity. I'm going to suggest getting multiple consumer-grade-ish brew machines, possibly with some fancier options (e.g. programmable turn on and off times). Since these features are so general, I cannot recommend a specific machine, but you can easily find some based on those criteria.
Info on commercial grade
Seattle Coffee Gear has a playlist on YouTube dedicated to coffee makers for commercial use. With your description, it's hard to give good advice, but I think you can get pretty far by going through all the videos (I am not affiliated with them, I just know they make good videos).
I suggest watching those videos to either settle on one of those options or to refine your criteria and post them here. Things you may want to specify:
Comment on cleaning and maintenance
As for cleaning, you're going to have to do some level of cleaning with any coffee machine. Can you minimise or automatise it? Yes, but never forget that it's there.
Also note that machines do sometimes break down or go out of the running because of maintenance. The best way to avoid that impacting your workflow too much would be to have multiple smaller machines (though that's not your question now).
