"Brake Cleaner" and "Brake Fluid" are not one and the same. I'm sure Hirlau is aware of that, I'm just pointing it out for anyone who might *skim* through this thread.
As for using Brake Cleaner.... yes, it's going to degrease and clean leather, just as it would any parts in a braking system on a car. Lighter fluid is NOT the same, it is a very very light distillate of crude oil, and in fact it IS an oil of sorts, and it would not make a substitute for brake cleaner when working on cars. Given my druthers, I'd reach for the lighter fluid first and in fact I just did, cleaning the gunk off a powered strop I use for sharpening knives and such (never a straight razor, or course). It works great and is not as drying as brake cleaner would be.
Now, I don't think using Brake Cleaner is going to immediately ruin a strop, but I would feel the need to replace some of the natural oils after cleaning.
As with everything else, we all have a pet ways of doing things.... I tend to reach for a waterless mechanics hand cleaner with lanolin when I try to revive an old strop (NO PUMICE) and I do not use the citrus based ones either, just the mineral spirits type (I think mineral spirits and lighter fluid are pretty close to each other in the grand scheme of things).
But, all the above aside, I'll try a damp microfiber cloth first (cotton terry if I don't have micro fiber handy) first... then go from there. Pears bar soap works pretty well for me, if I need to 'up the cleaning' a bit, it's much like the saddle soap I used when I worked at Assiniboia Downs as a swing groom. There was lots of tack to take care of, and much of that tack was 20 or 30 years old and older. It would get muddy, sand and grit ground in from the track, sopping wet from when it rained.... lots and lots of abuse, but a pure glycerin soap and some neatsfoot oil is all we needed to use. Sure, it would eventually kill the leather, but daily use and weekly cleaning over 30 or 40 years is more abuse than a razor strop will see in a century of use. I would suggest that a lack of use is what's killing most strops.
That is my point of view... right or wrong.
Regards
Christian