BA's can do absolutely the same damage as Mech's. A micro heavy plus a PPC can do some heavy damage. All the BA has to do is run under your feet or ride the rear of a tank.
The damage a BA can do is negligible to heavies and assaults, which comprise the biggest portion of the battlefield after the first 20-30 minutes. Unless you let them sit there and plink at you for a couple of minutes, they aren't going to deal any decisive blows. They aren't a threat to ASF, tanks have heavy armor and should NEVER operate alone, light/medium Mechs can outrun BA and, Hollander notwithstanding, kill them quite easily most of the time.
Do you know how frustrating it is to have BA's running right behind your tank and you can't traverse your turret down to shoot them?
Tanks shouldn't be able to have an easy time killing BA IMHO. They're tiny and more maneuverable, it's their only real advantage against anything. Tanks have the biggest disadvantage against them - they are slow and less maneuverable than Mechs, and often focus only on one type of weapon; specialists by and large, and most of them fall into the role of AA, Mech-killer, or artillery. AA is excellent for killing BA - if they get on your tail, hit your boost and back up. Unless you're in a Huit you should be able to outpace them once they burn their JJ fuel once or twice, and even then you shouldn't be alone or on the front lines and your buddies should have an easy time picking the BA off. Same applies to Mech-killers and artillery, let your teammates do the work, you should never go Rambo in one. One asset should never be Rock, Paper, and Scissors.
Weapons need to have more splash damage radius. It's frustrating that nothing but direct hits, PPCS, and SRMs kill BA's.
Listen to what you're saying. What weapons are these that you allude to? AC2s kill BA in 2-3 hits, and adding splash to them would be ridiculous due to their size and rate of fire. Same deal with RACs and LBX, though RACs they can be hard to aim with precision at close range. That leaves LRM/MRMs, other ACs, Gauss rifles, and lasers.
Lasers will never have splash damage, that should be self-explanatory, and even small lasers are deadly to BA.
LRMs really shouldn't be good at killing BA... but they are anyway. Try aiming down, you eventually get a knack for it. MRMs are better at it than either LRMs or SRMs, since there's almost always more MRMs on same weight class Mechs than SRMs, and they don't have the arc of LRMs.
AC10/20s and Gauss rifles are bad at killing BA, but not impossible to use for such. Assuming AC10/20 are similar to a HEAP tank shell, a tiny bit of splash damage makes sense (and IIRC they have it) but still, using one on a BA is like nuking an anthill and most Mechs have backup weapons that work better. Gauss rifles do work with lucky shots or accuracy, but there's really no reason why they should be reliably useful against BA given that they have superior accuracy, damage, range, and heat to PPCs.
MGs aren't that great at killing BA, but they work, then again MGs are pretty much just there to take up free pods and tons. Flamers really aren't good for anything except overheating energy boats, though. Any decently skilled BA can dodge them, and I have mixed feelings about this. On one hand, overheating is a really cool niche. On the other, they probably should be good at swatting BA as there really isn't a weapon dedicated to that role except maybe AC2s (again, lol @ MGs).
That leaves, what... ArrowIV, LT, Firebombs, and Thunderbolts? I'm not spelling those out. As someone who's logged hours as BA I hope I've shown you its perspective when faced with just about every weapon type. The same principle that applies to assets not being jacks of all trades goes for weapons. If they were all good against everything, the game would be no fun. Imagine only being able to use the Novacat. Might as well go play Modern Borefare 2.