I like Bowser's chances as a solo main because he doesn't have to approach matchups very differently. No matter who you are, you can't afford to get grabbed by Bowser, you have to respect OoS Fortress, you can't let him get too much rage, and Jab is a scary move to get around in the neutral. And when it comes to killing him, a lot of characters start getting Melee Marth syndrome, most prominently sheik, who has no kill setups past a certain point of damage on Bowser. Consider instead Ness as a solo main. Ness is probably as high as two tiers above Bowser, but he has an almost unwinnable matchup against Rosalina. Any time he's forced to recover with Up B, she can take it away from him. A lot of projectile oriented characters have unwinnable or absurdly difficult matchups against her, Villager, and sometimes G&W, they need secondaries. Bowser isn't in this position against anybody. ZSS, our worst matchup, is tough because of the same reasons it's tough for anybody. Her neutral is scary, her combos and kill confirms even more so, and she's not easy to edgeguard or read without putting yourself at risk.
If you're serious about competitive Smash 4, I would always recommend secondaries, but I would never say it's necessary for Bowser, not now.