For live mixers, yes, because if you turn down the lead vocalist you don’t want the reverb mix balance to change as the voice changes in level - you want the whole signal as a set to change. Otherwise if you wanted to turn them down or up, you’d have to simultaneously change the ratio of send, and that’s not really feasible when dealing with lots of signals at once in the middle of a show. Usually the types of effects on the aux sends never need to get even close to 100% wet (reverbs in live contexts are often quite minimal, compression is sometimes used in a parallel “upwards” style to prevent dropouts and equalize the occasional moments when the mic distance is uncontrolled, delays are usually not heavily prominent (the venues often have excessive reverb and untuned delay anyway, so adding much more is almost never a good thing for clarity). And nearly all small mixers (except perhaps some boutique offerings) are largely oriented towards gigging band live use as much or more than studio utilization.
For signals which need to go to 100% wet, you’d traditionally use inserts, and for groups which share the same 100% signal at the full mix (e.g. drum bus compression) you’d use an insert on a bus, and have the channels send to the bus instead of the master.
Studio usage is not radically different - each channel of a mixer is really thought of in general isolation from an effects standpoint and the inserts are used heavily to customize each channel’s effects set - in companion with a sideboard patchbay which has all insert sends and returns on one level and most of the studio’s effects patched into a second adjacent panel for rapid connection in series through any insert chain.
For flexible usage, as @madeofoak mentioned, some mixers offer the option to “flip” the source position of the send on certain auxes from post-fader to pre-fader for monitor usage (musicians always hear themselves at the same volume regardless of the house mix) and this makes occasionally for some creative utility when using send effects like 'verb and delay, but it’s not really a replacement for an insert, because you still have the dry signal going through the master fader and ending up “somewhere” that needs to be either muted or routed appropriately.
Mixers are not really intended for dynamic “routing” of multiple channels through multiple effects in any combination - for that you’d want a matrix mixer if you need more than the on/off routing of a patchbay.
Yes, most patchbays offer this functionality, even the cheap (and effective Neutrik 48-point 1/4" ones you find at any music store or online - I have one, I’m quite happy with it). Learn about “normalization” and the opportunities this offers for flexible routing. I have mine set up so that the signal from my sources runs in the top rear port, is default normalled to both the bottom rear port AND the top front port, and I have the recording interface (or mixer in your case) coming out the bottom rear. So by default the signal flows directly through to the recording interface and a copy of it is available at the top front port. If I plug a plug into the bottom front port, it breaks this normalling and connects directly to the recording input, so I can override the default source (which is still available at the front). This lets me insert an effect in series with the signal to the interface, or replace the signal that the interface sees with another entirely different one, as I have need from moment to moment.
It’s important to note that patchbays are passive - connecting a source to multiple destinations at the same time can change the impedance the source sees and depending on the specific electrical characteristics of both the source and the connected “listeners” the frequency response and level may chance in significant and unexpected ways (e.g. you may get rolloff or filtering, you may see that one source sees a much higher or lower level than the other, or both sources may see a much lower level than one source alone would see, among other possibilities). You also can NOT use the patchbay as a mixer (e.g. using this combination to combine two or more sources into one send).
