4/4 would give a completley different feel, there are no groups of three in 4/4.
...A triplet played in 4/4 can be a group of 3 per beat...so yes...there CAN be groups of 3 in a 4/4 timing.
Remember that time signature does not describe the music...it describes how to commonly describe the music...and the time signature given to ANY music is arbitrary...and intended for communication purposes.(one persons 4/4 is anothers 2/4...or 5/4 that finds a common "1" every 20 beats...depending on the musical phrase described, of course)
The descriptive charactaristic of "time signature" is ARBITRARY.
You can play something noted in 4/4 with a "3" feel.
You can play something written in 6/8 with a rigid "2" feel.
Of course, both are ruled by the accents and additional charactaristics of the written music...such as ties and even hand written notes in the margin from the writer.
A time signature is not music...it is part of an arbitrary set of conventions to describe music. Western Notation gives us infinite ways to describe one specific musical moment.
disclaimer: I only call myself professional when I accept money...so please feel free to ignore me ; )