That said, I did some experimenting today. I wrote a computer program to simulate rolling contests vs rolling against a static DC. The differences are pretty minor and boil down to your bonus vs. your opponent's bonus (or the DC difference from 10 for a static DC). Unless your opponent's bonus is 5 or more higher than yours (or the DC is equal to or higher than 15+your bonus) you are slightly better off rolling against a static DC by a small percentage. At an extreme range of +-10, the difference my be as high as 15%, but normally it is much lower.
I think my only gripe with passive skills is that they are based on 10, making them almost as good as an average roll of an active attempt. I think 8 would have been a better base, but 10 was likely chosen for simplicity of math.
It works fine for one on one like that. If you have a large enough number of individuals & roll for them all them one of the seekers will get a 20 & one of the hiders will get a 1 and it's very different compared to just rolling one side (the PCs or the active party).
Passive being based on 10 is for when you might ask for a roll, or roll on the players behalf, in earlier editions. The benefit of actively searching is that it is
in addition to checks against passive values & is unbounded upwards too.
Rolling both sides makes it just about impossible for large groups to hide from one another & the fact that this does happen in real life (eg the battles of The Trebia or Omdurman) makes it a bit jarring. In fact I prefer one side rolling because it makes this sort of situation more possible rather than effectively giving a -19 to hide checks. It still clunks for groups much larger than 4.
This is one of the reasons for using group checks as described in the DMG - it allows the rules to work in situations where they otherwise might give the sort of results you do not like. Group checks can do this the other way around - they are often a bit too easy & can be much easier with 4 than 5 as the maths is a bit odd.
They are all just tools to use to create the game you like so I would try them & see what feels good for you.
(apart from the static passive vs static trap DC issue - urrrghh!!) .
That is indeed an abomination.
I might pre-roll traps & write down the passive DC rather than assuming a 10 & that would be fine in a pre-written module too.
My motto as you will have figured out is that one side rolls, but then I feel many DMs make/demand too many rolls.
Many people like more randomness but I prefer chess to slot machines*
*This is a lie I hate them both