I understand that the aircraft can pitch up faster than it will
What you are describing here, is an increase in angle-of-attack. Over any given time interval, if the pitch attitude has increased more than the climb angle has increased, this generally indicates that the pilot has moved the control stick or yoke aft to increase the wing's angle-of-attack.
During aerobatic maneuvering, the angle-of-attack will often be increased, but normally not extremely rapidly, and normally not to the point of reaching the stall angle-of-attack-- with the notable exception of snap rolls, spins, etc. For non-stalled aerobatic maneuvers, variations in angle-of-attack are typically in the neighborhood of 10 degrees or less, while variations in the pitch attitude may be unlimited.
Note that if we start from straight-and-level flight, a very rapid, large increase in angle-of-attack will invariably impose a very large G-load (lift force) on the aircraft during the portion of the maneuver before the stall angle-of-attack is reached, because the airspeed will not have time to bleed down to a lower value appropriate to the new, increased A-o-A. The need to limit the G-load (lift force) to safe values, typically limits how rapidly we can safely increase the angle-of-attack.
So normally during aerobatic maneuvering, we do not "pitch up faster than" we can "actually climb" to any significant degree, except for maneuvers involving intentional stalls.
Your model of a pilot commanding pitch attitude changes which can't be adequately matched by changes in the direction of the flight path, resulting in very large changes in angle-of-attack, doesn't accurately reflect how aircraft are actually flown in non-stalled aerobatic maneuvers.
We've noted above that the rate at which we can safely increase the aircraft's angle-of-attack is constrained by the need to keep the G-loading within safe limits. But this inherent limit on the rate of change of the angle-of-attack does not equate to a limit on the rate of change of the climb angle or pitch attitude. During aerobatic maneuvering, the pitch attitude will typically change much more rapidly than the angle-of-attack.
Even in a low-powered or un-powered aircraft (e.g. sailplane, hang glider) it is quite possible for the direction of the flight path to pass through vertical into inverted flight, without the wing ever reaching the stall angle-of-attack.
In fact, it would hypothetically be possible to fly some sort of modified variation of a loop with the angle-of-attack remaining constant throughout the maneuver. (The aircraft would have to be quite strong, as the G-load (lift force) at the beginning of the maneuver, when the airspeed is high, would be very high.)
but at some point, the elevator will lose control authority until the
flight path gets closer to the pitch angle
I'd suggest that this concept doesn't really make sense. Yes, if you increase the angle-of-attack all the way to the point of the stall, control authority is affected somewhat, but normally the elevator continues to work. (Granted, in cases of extreme post-stall maneuvering, where the angle-of-attack is far beyond stall and the aircraft is being controlled by thrust vectoring etc, the elevator may be ineffective, but that's outside of the available flight envelope of most of aircraft.) As explained above, you seem to be envisioning a very rapid increase in angle-of-attack. Note that during the (initial) portion of this maneuver where the wing is not stalled, this will impose a very high G-load (lift force), which will cause a rapid upward curvature of the flight path.
So basically it seems you are describing a poorly-flown entry into a loop, where the flight path initially curves upward but at some point the pilot applies too much back pressure and stalls the wing.
Fundamentally, you seem to be viewing the elevator as a pitch-attitude control, and you are recognizing that this eventually leads to changes in the direction of the flight path ("until the flight path gets closer to the pitch angle"). I'd suggest that you think of the elevator as an angle-of-attack control, and recognize that pitch attitude changes are the combined result of changes in the angle-of-attack and changes in the direction of the flight path. A rapid change in pitch attitude is not necessarily an indication that the angle-of-attack is rapidly increasing, i.e. that the rate of change of the direction of the flight path is somehow "lagging" behind the rate of change of pitch attitude.
For example, we could demonstrate a rapid increase in pitch attitude even in a completely unpowered aircraft, without even touching the control stick at all. In such a maneuver there would be no reason to suspect that the angle-of-attack was increasing at all. Example-- in a glider (sailplane), trim the elevator for flight at the best-glide airspeed and angle-of-attack, with the wings level. Remove your hands from the control stick. Use the rudder to slowly roll into a steep turn, which will cause the airspeed to slowly and smoothly rise. Once the airspeed has stabilized at the higher value appropriate for the steep bank angle, use the rudder to briskly roll to wings-level, applying opposite rudder as needed to stop the roll and "freeze" the bank angle near horizontal. Due to the retained excess airspeed-- which equates directly to excess lift-- the flight path will curve upward sharply -- but to a first approximation the angle-of-attack will remain constant throughout the maneuver.
As other answers have pointed out, with sufficient airspeed and appropriate control inputs, there's no issue with bringing the flight path all the way to the vertical for short periods of time, even in unpowered aircraft (e.g. sailplanes, hang gliders.) Loops are routinely flown in such aircraft.
Perhaps some aircraft could use a combination of maximum elevator and
an airbrake to achieve extreme pitch rotations
If the aircraft is performing a "Cobra" maneuver where the pitch attitude is brought to (and beyond) vertical with little change in the direction of the flight path, then it's difficult to say how various airbrakes, spoilers, etc might help or hurt the maneuvering capability in any given aircraft. However if the pitch attitude is being brought to vertical in the more conventional way-- by curving the flight path through vertical-- then according to Newton's laws, a force must be applied to make the flight path curve or bend -- because a change in the direction of the flight path is a form of acceleration. And that force is Lift. If the airbrake or spoiler is located in a position where it disturbs the airflow over or under the wing, then that will reduce the lift coefficient. That's not going to help the pilot "pull G's" (apply extra lift force) and curve the flight path.
Note that at normal airspeeds, pitching maneuvers are not usually constrained by the total amount of pitch torque that is available. At normal airspeeds, the pilot usually has plenty of control authority to put the wing at any desired angle-of-attack, all the way up to the stall angle-of-attack, just using the elevator alone.
Rather, with a low-powered airplane or glider, the issue is simply making sure that the aircraft has sufficient initial airspeed so that the airspeed will not drop to an unacceptable level-- meaning that the wing's lift force drops to an inadequate level-- before the maneuver is complete.