That is true. If you go that route, I'd recommend you aim for Fighter 1/ Bard 3 first. With advantage and one attack, you can at least grapple and throw opponents off of cliffs or take two rounds to knock them prone. At Fighter 2 / Bard 3 you unlock Action Surge, which allows you to lock down one opponent in one round at least once in a while (and combos really well with Earth Tremor). Then maybe go Fighter 4 / Bard 3 to pick up Tavern Brawler so you can pull off the one attack - trip - bonus action grapple combo. That's only 1 level later than a pure bard or barbarian/rogue would be in getting the ability to grapple and knock down an opponent in one turn.
Ah good call. Still, I think I'm leaning more toward the Barbarian/Rogue route. The earlier effectiveness and the jump/falling damage attack are what sold me (prone with movement, and still have actions/bonus actions? Heck yes). Plus no need to bump Cha is nice, and raging/sneaking fit the bugbear better than strategizing/barding. Though I still may pick up Fighter 3 later in my progression; seems like high-level Barb and Rogue stuff isn't quite as useful to a grappler once I get the basics down.
One question on that barb/rogue build - since you need a finesse weapon if you want to get sneak attack bonus damage, have you had any luck with coming up with DM-acceptable finesse improvised weapons? I think iron spikes as improvised daggers should be an easy sell. I'm also hoping to use a roasting skewer (like you'd spit-roast a chicken, or small pig, or gnome on) as an improvised rapier, and/or a rope with a small spike or caltrop tied to the end as an improvised whip (or maybe even length of chain). But I realize those may be tougher to convince a DM on. I play AL, so I'm expecting DM variation.
(For levels 4 and 5 though, while I have Tavern Brawler but no Sneak Attack to worry about, I'm going with a tree branch with various sharp pieces of iron driven through it as an improvised --and very bugbear-thematic-- maul.)