You are right about the "broom and shovel" method of clearing scarification and coring debris being slow and labour-intensive. I would normally recommend that the means of material collection and removal be considered when you purchase scarifying and coring machinery.
Let's deal with the scarification material first. Many walk-behind scarifiers offer some form of boxed collector - though the amount of material normally means frequent emptying of any collection system is a necessity. You need barrows or trailers standing by.
Another solution is to let the debris lie on the surface and then run over the area with a vacuum collector - something like the Billy Goat or Bear Cat leaf collector. Or you could use a handheld leaf blower to gently nudge the debris into one corner and then pick it up from the pile. Using a blower for such lightweight material does take practice and a little bit of skill - not too many revs.
For large areas such as pitches, I suggest you look at mowing/scarifying/collection equipment such as that supplied by Trilo, Major and Amazone, although suitability will largely depend on your need to use a flail mower. Etesia offers a scarification implement for use with its ride-on Hydro 124 rotary mower.
Hollow cores clearly weigh that bit more than scarification debris. And if you are not quick enough, they can dry and then have an annoying tendency to break up, making removal even more tedious.
There are a couple of machines available specifically for core collection. Charterhouse Turf Machinery offers the CC3000HL. This pedestrian unit has a 1cu m capacity hopper that tips hydraulically for a height of 1.3m for emptying into trailers and skips. It is powered by a 7.5hp B&S petrol engine and has six forward speeds plus a reverse. The collection head is designed to float over the undulations on golf greens.
Turfmech Machinery of Staffordshire makes the SQRL and CAT 800 core collectors. The SQRL is a pedestrian machine with a 60cm wide, fully floating rotary brush/paddle collection-head designed to follow contours for one-pass collection in wet or dry conditions. Sweeping height and brush aggression are adjustable to suit the type and density of material being collected. Power is supplied by a 13hp Honda engine and the variable forward speed will go up to 7km/h. There is also a reverse.
The CAT800 is a ride-on machine. It's a three-wheeler, 80cm and again has a fully floating brush/paddle. With 20hp low-noise Honda engine and a high-dump 0.4cu m hopper, this machine will travel at speeds up to 14.5km/h.
Although the core collectors are an additional cost to the coring operation, they can also be used to collect scarification material.
Depending on the hollow corer you use, there may be attachments to help. It's worth contacting your dealer to find out. Groundsman Industries' tractor-mounted hollow-coring units, for instance, can be used with its core collector for one-pass operations. Many aerators are also offered with optional windrowing equipment.
Sally Drury has reported for HW and its forerunner GC&HTJ for 25 years, and has spent more than five years testing machinery for HW and What Kit? The advice in this helpline is independent.