Why a plain random shuffle is not enough
Spreadsheet shuffles and name-picker wheels do produce a random arrangement, and about one time in five they produce the arrangement you specifically did not want. That is the nature of randomness: it has no idea that two of the names should never appear together. You end up rerunning the shuffle until it happens to give an acceptable result, which is neither random nor efficient.
Here the constraints are part of the process rather than something you check afterwards. Auto-arrange places the most constrained people first and only then fills the remaining seats, so every result it produces is already valid. Press it as many times as you like; the arrangement changes, the rules hold.
Good moments to reshuffle
In a classroom, the usual triggers are the start of a term, the beginning of a new unit, or the point where a group has stopped working. Regular reshuffles also make the seating plan feel like routine rather than a punishment aimed at one table, which tends to reduce the amount of negotiation you get about it.
At events, reshuffling is useful for workshops and training days where you want people to work with different colleagues in each session. Build the list once, then reshuffle between sessions and print each round.
Keeping some people fixed
Auto-arrange reassigns everybody, so if some placements must not move — a student who needs to be at the front, a speaker who needs an aisle seat — the practical approach is to run the shuffle first and then move those few people into position by hand. With a typical class or event list that is under a minute of work.
Where this fits with the other tools
If you are planning rather than shuffling, the classroom seating chart maker and the wedding seating chart maker use the same engine with wording and defaults suited to those situations. All three save separately in your browser, so a class plan and an event plan will not overwrite each other.