Frame and fork material
For a gravel bike without suspension, which is most current ones, a lot of the suspension and vibration damping comes from the tires. Beyond those, all frame materials can be tuned to offer more or less vibration damping. Carbon is more tunable than metals because it's not isotropic, i.e. it can be made so that its stiffness is not identical in every direction. However, that doesn't necessarily mean carbon has better damping than metal - that depends on the layup of the fibers and the fiber types selected. You could make a carbon frame ride harsher than a steel one, if you chose to do so.
Titanium and steel are considered to have a good ride quality, at least in the tubes that are commercially available. At least part of this is likely subjective, and some riders may perceive steel or titanium as riding more pleasantly than carbon. NB: I have a somewhat classically styled steel road bike, and a carbon gravel bike.
It is not really easy to compare the ride characteristics of the Bearclaw to more mainstream carbon bikes. The carbon bikes' ride characteristics will vary depending on what they were designed for. You would have to seek out reviews by competent and experienced reviewers. I generally recommend the crowd at Cyclingtips. For example, they assessed the Evil Chamois Hagar's (carbon) ride quality as poor, even "backbreaking", and the race-oriented Cervelo Aspero as similarly uncomfortable, saying that it was "brutal" on rough terrain. In contrast, they felt that the Santa Cruz Stigmata was much more comfortable.
I am not aware of reviews for the Bearclaw, but I'd caution that just because its titanium may not mean that it rides like other titanium bikes. While metals are isotropic and can't be tuned through layup like carbon fiber, a manufacturer can tune the ride by selecting different tubing diameters, thicknesses, and shaping. I suspect that the ride quality between titanium bikes is pretty heterogeneous. It's probably more homogeneous than the ride between different carbon bikes, but the ride quality alone should be able to vary substantially, as well as the bike's handling (due to geometry). That said, the sorts of people buying titanium or steel bikes tend to be seeking comfortable rides.
As to forks, as far as I know, titanium forks are relatively uncommon. As you mention, Bearclaw seems to offer a full titanium fork, and they may have done so for some time. Because of the variety of carbon forks available, and the fact that titanium forks are likely heavier than carbon for the same stiffness, I suspect titanium forks simply haven't taken off commercially. One possible rationale for a steel or titanium fork from a custom builder would be that they can select their own fork rake and length; otherwise, manufacturers will commonly spec one or two rakes across the full spectrum of frame sizes, which does mean that the way the bike handles may change with size. I'd expect a titanium fork to be good at vibration damping, but I doubt it's necessarily superior to a well-selected carbon one on an off the rack bike. Bearclaw appears to offer off the rack only.
Suspension is still relatively rare on gravel bikes. Potentially, it can do two things.
More prominently on gravel bikes, it can isolate the rider from chatter. This reduces fatigue. Some bikes have front suspension only, like the Specialized Diverge. Generally, gravel bikes with built in suspension offer a lot less travel than mountain bikes. You could also put suspension in components, like Redshift's suspension stem and seatpost, or Cane Creek's suspension post.
Suspension forks and frames keep the wheel in contact with the ground over large bumps. This is important for mountain bikes. I am only aware of one current full suspension gravel bike, the Niner MCR. There are aftermarket suspension forks, e.g. the Lauf Grit, which I don't believe needs a frame with suspension-corrected geometry to mount. However, I'm not sure how much gravel cyclists need this type of suspension.
Here is a Cyclingtips review of the current Specialized Diverge, which as noted has front suspension. Testers rated it as very comfortable to ride. The review of the Niner MCR full suspension bike is also by Cyclingtips. The reviewer noted that the improvement in traction was tangible.
All else equal, I'd actually expect suspension systems to reduce efficiency, as they can bob as you pedal. One can tune the suspension (as I understand the issue, not having had much experience on mountain bikes) to minimize this. That is one minor downside: if you have suspension, you will need to tune it to your weight to achieve optimal performance. However, the more complex suspension systems do have greater maintenance needs. The BMC Urs has a softtail-like rear suspension that should require periodic cleaning and regreasing of the pins and bushings, and maybe occasional replacement of the elastomer damper. The Niner MCR's full suspension system should be much more complex to maintain. Interestingly, Specialized claims the Future Shock is fully sealed and doesn't require maintenance.
There's another complexity issue if we are talking about full suspension bikes. As a traditional suspension fork like the one mounted on the Niner compresses, your head angle steepens. The Cyclingtips team noted that the Niner's handling felt odd, "like a 29er mountain bike from yesteryear". Bike manufacturers looking to integrate more complex suspension systems will need to think about how the suspension interacts with the type of handling gravel cyclists are looking for, which tends to be different from what mountain bikers are looking for (although not exclusively; some gravel bikes like the Chamois Hagar, the Urs, and the Diverge borrow a lot of influence from MTB geometry, although they aren't identical).
Do note that not all front suspension systems cause the head angle to steepen as they compress; the Diverge's doesn't. The tradeoff is that the Future Shock is limited to 20mm travel. This would be clearly inadequate on a mountain bike, but the Diverge isn't a mountain bike.
It's worth noting that suspension is still rare on gravel bikes. I've written this assuming you were talking about a gravel bike with a suspension system more significant than just a seatpost and stem. As indicated, I'm currently aware of only one full suspension gravel bike. Current gravel bikes are trending towards as much as 700x45-50mm tire clearance, which is a lot of tire with which to cushion the ride. I'm not sure how much suspension is actually necessary.