Используя Скрученный по сравнению с Твердым CAT5 в кабельном желобе

Use solid core wire for structured wiring (in-wall, above ceiling, etc), it's cheaper.

Use stranded core wire for everything else (eg, wall to computer, in-rack, patch panel to switch, etc)

To directly address the question as stated:

I remember back in the day it was a big deal to use solid cable for horizontal runs

That makes no sense at all - whoever made this a big deal was uninformed at best.

that was usually between a switches or other stationary devices

As stated above, solid core is for structured wiring, not "stationary devices" which have a tendency of moving over time. It should only be used where the endpoints are part of the building.

I've been seeing problems with similar situations where a solid cable was used because people move stuff around and eventually the conductors break.

Correct - which is why you never use solid core outside structured wiring.

Now in the modern day does anyone see a problem with using stranded cables for this type of installation?

You never cited any actual problems with using stranded core cabling in the first place. The main reason it wasn't used for structured wiring is price, stranded is more expensive than solid core. A minor secondary reason is that solid core is much easier to get a good connection in punchdown blocks and keystone jacks.

Note: There are different C8P8/RJ45 ends for Solid and Stranded wire. Be sure you use the correct end for the wire if you want a reliable connection. The pins when viewed cable end-on look straight for stranded wire, and look forked for solid wire. Diagram courtesy cableorganizer.com.