You can train under more open rulesets while still having only a few go to tech for that ruleset. An important aspect of training under more open ruleset is learn to defend the different attacks that can come at you.
Boxers have quick reactions under their own rulesets. The learn to see punches coming. How they be able to dodge or defend kicks? Or what if you can actively clinch and attack? With clinching I bet their reactions get slower.
I would say yes, and no. One of the things I've noticed in other disciplines specifically MMA is that their fighters seem "flinchy" at times. One crucial element in boxing is learning to keep your eyes open. The punch that hurts the most is the punch you don't see. That flinch reflex is for the most part trained out of practicing boxers.
Take Floyd for instance, have you ever looked at his eyes when he fights? They are generally wide like an owl. It almost looks comical. The eyes provide sensory information. Boxers learn to train their eyes, and quickly. Punches are faster than kicks, as arms are closer to your face and torso. Low kicks are a different story because you're taking the fight downstairs.
As far as training for kicks, yes, you'll have to train for those and be aware how they're coming at you. Personal example (not the standard--take it as you will), I've sparred with karate guys and MT guys. My reflexes and speed was greater than theirs. In boxing you have to transition from being relaxed to being tense in a rapid manner. This is why they're able to generate such power at close ranges (tight hooks/tight uppercuts) that knock opponents out. Their reflexes allow them to accelerate their mass rapidly over a short distance to generate sufficient power.
Now could they attack areas that I don't have regularly covered? Absolutely. However, anything aimed at my torso or head in my experience was easily dealt with through reflexes, guard changes, and pivots once I settled into the rules and understood the direction the attack was coming from. The majority of their success in kicks came from attacking low when my stance was wide. A more narrow, mobile stance was a different story. However, I work footwork more than I do actual punching because I do not enjoy being hit. I will go entire rounds just pivoting, back pedaling, shuffling and walking off. No knock to any style, just my experience and observation.
As far as clinching, that is already in the sport. Clinching is grappling so the speed difference would presumably be there from that alone. Being able to clinch in the match is dependent on the ref. Boxers actually have solid clinch work. Sparring there is no ref to keep it honest, and they can turn into high level brawls which inevitably involve clinch fighting. At the boxing gym I go to we actually go whole rounds where we just work clinching and off balancing, collar ties, over hooks, under hooks, etc.
In conclusion, I believe boxing has great transfer for reflex training.