it would not have been impossible its done all the time
the use of 45 and 90's degree fittings. i suggest that you open the walls up and run a vent line. what you have right now will not work. no vent..
i will explain a vent...take your glass of coke,put a straw into the glass. put your finger over the straw and pick the straw up..it does not drain..no vent...now..take your finger off the straw VENT.
there is one option..before i tell you
can you tell me where in the cabinet,the pipe is? and what size it is? also..is your basement under the sink?

They're not code because the water drains faster through them, drawing extra fluid out of the trap as it goes (potentially breaking the seal).

That being said, our house that has an S trap in the kitchen. No problems at all.

Other family members have them in another older house - again, no problems at all. Drain fast and fine, and yet no indication the seal was broken (no odors).

your correct homebound, there are sinks all over the place with the old style plumbing in them. but..we have a code
and if the op will be willing, i will be willing to get his sink up to code. an s trap is illegal because as the water leaves it, it causes a condition called back syphonage
it pulls all the water out of the trap, leaving a dry trap
then the stink monster enters the house and funks it all up
making momma bear all grumpy...LOL...
the stink is also methaine gas, which is flamable..
not a good thing for a kitchen with a cook top ...
it won't happen the first time, maybe not the 2,3,5,or the 69th..but MAN..i do NOT wanna be around when it does
true story....1984..jackson mississippi..terry road..
a plumber opened a man hole lid, then lit a cigerate
the hole went whoosh and 200' away another man hole cover
flew up 5' in the air..the man was my boss. i was 20' away at the truck it was an AWSOME sound

I actually have two sinks with this problem, the main sink and a smaller bar sink. Both drains are 1 1/2" and located on the main floor over the basement (which has a finished drywall ceiling). In both cases, the drain is located off to one side of the cabinet, midway from front to back. I understand venting.

One sink is located only about 3 feet from the stack, so it is properly vented. The other sink is located across the house from the stack, and I was unable to route a vent, so I planned to install an air admittance valve for that sink.

I have 2 situations that are likely considered S-traps. Yes, another DIY blunder. But let me splain, first? First one is a proposed floor drain to be constructed just like a shower floor using a 3-pc floor drain except it will be used as a washer/dryer closet. I got concerned about it leaking w nowhere to go. Anyway, to get it to connect to the main 4" drain running horizontally, the cramped space between floor joists offered few options, so P-trap ties directly into a 4x4x2 sanitary T. This floor drain may be used once a year. Second is a standpipe for the same washer. Again, the P-trap ties directly to a 2x2x2 sanitary T out the top of which is the vent. Am I another violator of every plumbing code in canada and the US?