Romance, or Jeux Interdits, is DEFINITELY a piece well-known in the classical guitar repertoire, regardless genres and abilities to play the guitar. The popularity of the piece is probably due to the simple melodies, while maintaining technical interest if one plays well enough (those arpeggios look cool). However, it can backfire the player if they overplay this piece, and not for technical and musical development sake, ultimately resuting in boredom, and even burnout in playing the guitar.
Apart from Romance, there are pieces that are mainstream, played by almost every guitar player, however with little musicality and technical refinement. These include: Canon in D (J. Pachelbel), Lagrima (F. Tarrega), or some of the starting chords from Concieto de Aranjuez (J. Rodrigo). Though there is nothing wrong with these pieces if they are played in full with exceptional musical capabilities, repetition in the beginners' guitar world gradually results in the lost values of these "supposed to be famous" works.
La Catedral (A. Barrios). Yes, and just need to play the first two movements. Although it can be a bit overwhelmed to someone new to classical guitar, the technial challenges are perfect for skill development. Also, the nostalgic melodies, plus the history of the piece is nothing to be missed. This piece can replace Romance and Lagrima.
Prelude from BWV 1007 (J. S. Bach). Although many people, mostly post-amateurs, argue that this is also one of the most overplayed piece, I think it still has the value on the guitar (not the cello). Also, learning this piece can also familiarize yourself with the music of J. S. Bach, knowing how hard it is to play an easy piece. This is a suitable replacement for Canon in D.
Cavatina (S. Myers). This piece might not be in everyone's YouTube recommendations, but it has a jazzy feel to it which is perfect for a rainy day. There are some big strecth to be careful of, but overall a really nice and chill piece to play. Play this instead of Romance.
Romanza from Grand Sonata (N. Paganini). Despite sharing the name with the Romance, this piece is definitely at a whole new level, while being playable at beginners level. This piece can be a showstopper if played extended musicality, showing the feels of the piece.
Slow movements from the Sonatas for Violin BWV 1001, 1003 & 1005 (J. S. Bach). Another work by Bach, however now with a different style. Just play from the original scores for the violin, and it works surprisingly well with the guitar.
I specifically chose those pieces since they offer similar technical demands (more or less), while having high musical value. This is not to say Romance does not have any musical value at all, just that the piece is played again and again, making it it boring to listen to. In other words, playing these new pieces gives you, and your audience a fresh new look of the classical guitar repertoire.