The I, IV, and V chords will certainly be enough to make a progression that subjectively sounds like it 'works', by most normal people's standards, under any melody that genuinely restricts itself to the major scale. As you say, you have all 7 notes of the key contained within your chord options (which allows any note in the melody to be a chord tone if you wish) and the transitions between all of those chords will sound very standard and subjectively acceptable to most.
As to whether it's possible to always make something that sounds good is more subjective - you might always be able to make something someone would like - but I think you might be in danger of boring yourself by using such a restricted palette of harmony.
I ask this because as I'm learning songs I know in different keys
It sounds a bit strange to be asking about which chords to use if you're learning songs you know. If you're learning songs you know, you (IMO) need to use the chords from the songs, or they won't sound right! You could restrict yourself to songs that do only use I, IV, V chords and play them in different keys though, if that's what you mean - in fact that sounds like a great exercise to me.