"As for the foundation drain - the builder called it a foundation drain

and hasn't mentioned anything about draintiles, but I'm not a
construction expert and am not even sure what is a drain tile"

You say you don't have a sump pump. So where does the foundation drain
go to? It has to take the water away somewhere.

As for thinking about applying a sealer to the floor and forgetting
about it, that isn't going to work. If there is water, it has to be
eliminated. Sealers almost never work as the water just finds another
way in. Or the pressure overcomes the sealer. That sealer the builder
applied that looks like caulk is useless.

I'd start by taking a look outside when there is a heavy rain. See
where water is actually going, pooling, etc. You may be suprised. For
example, I once had a problem I couldn't figure out. When I finally
went outside during a downpour, I realized the flexible 4 inch pipe I
had on the end of one of the leaders was backing up and water was
spewing out the top of it because it wasn't placed high enough up on
the leader.

Then I'd figure out where the foundation drains are going and get a
sump pump put in. There's nothing wrong with having a sump pump to
solve this problem. With a history of water on the floor, I'd most
certainly want one anyway to prevent a future possible disaster, even
if you don't need it to fix this problem.

Finally, in many cases these problems are worse with new construction
and may get better over time. That's because with disturbed earth
around the house, it's easier for water to get in. Also, if you want
to figure out where the water table is, you can get one of the round
auger type post hole diggers with a couple add on lengths of pipe.
Just did a hole and keep going till you find the water level.