Regarding the use of lawn clippings as mulch or compost, a reel mower simply clips of the top of the grass blades and spits them (if you choose) into the bag.
A rotary mower as commonly used does pretty much the same, although it may "bruise" the clippings a bit more.
A rotary mower with a proper mulching plug/deck traps the clipping in the deck for a couple of turns, and produces reasonably effective shredding so the bits are not so long as to be (as) unsightly if left on the lawn, and/or compost much more rapidly if that's your choice.
Unfortunately, there are few mowers with mulching decks/plugs that permit blowing the clippings directly into a bag or other receiver. Most assume that you're "mulching" because you intend to leave the clippings on the lawn.
If you wish to shred the clippings "properly" but still want to collect them for mulching elsewhere, or for composting, a handy device is offered in the US variously called a "lawn broom" or "lawn sweeper." Versions are available to tow behind your mower, and you can get powered/self-propelled ones; but since you'll use it immediately(?) after mowing, you'll have a level enough lawn to use a hand-pushed version reasonably easily.
Of course, re-mowing with your reel type mower, after chopping up the clippings with the mulching rotary mower, with a collector bag/box on and with the height adjusted close to the same height at which you'd just mowed, would be likely to do a pretty good job of "sweeping up" at least the larger of the clippings. Most of the sweepers closely resemble a reel type mower but with "slightly modified blades" for better(?) pickup.
(According to cultural conventions here, "He who dies with the most toys wins." This should be factored - especially - into the selection of lawn maintenance "tools.")