Does the first-move advantage for White have real meaning apart from the highest levels of play? Most games that involve taking turns, with one player going first by lot or agreement, have some sort of advantage to going first. Tic-tac-toe, for instance, is a guaranteed draw for player 1, assuming correct play. Connect 4 is a guaranteed win for player 1, assuming correct play. The first move in Go is considered so powerful that player 2 receives a bonus of 5.5-7.5 points. The first-move advantage in Chess is less obvious, although statistically speaking White has an advantage of ~52-56%.

White's advantage is essentially that of time. He can count on starting ahead of Black and in theory should not go behind except by choice. He can even attempt to trade material for time.

Black's advantage is that of choice, or response. She can let White develop the game as he chooses, or she can force the game into a different line of play altogether. Presumably she knows her purpose in playing her responses to White, and can take advantage of White's loss of choice.

This being said, does it have meaning outside the highest levels of play? Is the game theoretically close enough that there isn't a way to equalize it 100%? (Related previous discussion suggests that the first move is "worth 25-45 rating points". However, the question didn't directly deal with the first-move advantage, instead referring to it tangentially in a discussion of handicaps.)
I think any real answer to your question will have to be statistical in nature.  There's rationale behind the advantages and disadvantages of having the first move, but really, we'd be mostly guessing regarding how important these factors are.

With that in mind, I quickly ran some code to check what patterns I could pick up through the million base PGN library (check below for disclaimer / methodology):

ELO                  %W      %D      %B
1700: 56    games:  37.50 / 33.93 / 28.57
1800: 192   games:  34.38 / 35.42 / 30.21
1900: 736   games:  37.23 / 34.24 / 28.53
2000: 4682  games:  36.27 / 32.14 / 31.59
2100: 10568 games:  36.56 / 34.79 / 28.64
2200: 23486 games:  34.42 / 38.11 / 27.47
2300: 33444 games:  31.25 / 45.06 / 23.69
2400: 49706 games:  28.53 / 51.73 / 19.74
2500: 40264 games:  26.38 / 57.28 / 16.33
2600: 16946 games:  27.88 / 56.14 / 15.97
2700: 3581  games:  28.43 / 56.07 / 15.50
2800: 68    games:  27.94 / 55.88 / 16.18


As you can see, at the highest levels (around 2500 and above), the majority of games are draws (around 55%), and from the remainder, wins by White outnumber wins by Black with a ratio of around 1.75:1.  A very significant advantage, indeed.

The ratio drops rapidly at lower levels, and by the time we are around 2000 in rating, it's closer to 1.2:1.  From here, it doesn't seem to change much, and I don't really have data for under 1700 rating.  Of equal importance is the percentage of drawn games - along with the White / Black win ratio, the number of draws drops significantly.  I would interpret this data along the following lines (although I'm reading a bit into the data here):

White does have a tangible advantage, but needs to play very accurately to capitalise on it.  At the "lower levels", tactics (and thus mistakes) play a far larger role, almost drowning out the underlying small advantage.  It is still noticeable, however, even at the 1700 rating level, and with a small sample size.  So, to answer your question: yes, the choice of colour is relevant even at lower levels, although much less so than at GM level.

Disclaimers and whatnot - I ran through the million base PGN library (+- 1.7 million games), and simply discarded any games with a greater than 50 rating points difference between the players, in order to select only games between similar strength players.  Statistically, this shouldn't make all that much difference (the "unfair" games should have each other balanced out, given sufficient numbers), but there are a number of exhibition and simul games in that library which I was attempting to exclude from the data set.  In any case, this is not meant to be strictly scientifically correct; it's just the results of a few minutes of programming.