I rode with laced shoes, with no devices for keeping the laces tucked away, for many years. IIRC I frayed one carelessly tied lace in a lace/chainwheel/chain interaction. Was a little less sloppy after that, & it was never a problem again. Shortened over-long laces, tucked 'em into shoes, tied them to one side, etc.
I now have three pairs of cycling shoes: two with laces & a velcro strap to hold 'em down, & one (the newest) with just velcro. I prefer the laces + velcro combination. Better adjustment of tightness.
I still wear normal non-cycling laced shoes (i.e. no restraining devices) on my Brompton. I have never had any problem with laces snagging.