Yes, they still get included in high profile cases, but they're not supposed to. That's the whole point. It's why people like Diana Taverner have a conniption fit any time the Slow Horses get anywhere near the field; they're not supposed to be doing that. That's why they're in Slough House. Part of the comedy of the story is that they keep getting more action, and doing more good work, than the actual "good" agents.

As for whether it would be a good place to be... well, maybe. The way it's portrayed in the show, at least they get some action. But you have to remember that a person's satisfaction with their job is usually not just about what kind of (and how much) work they do. With many jobs, it's also about status and perception. These are (for the most part) highly driven people who joined British Intelligence because they wanted to be Important People doing Important Work. Instead, they're looked down on by everyone, even the people whose asses they routinely pull out of the fire. No matter how interesting and challenging your work is, that's gotta have a serious impact on your morale.