I disagree entirely. By giving Poison offensive capabilities, you give the Poison types that are, by their individual stat distributions, inclined towards offense more room to breath, whilst the bleeder strategy would still see use on the defensively inclined species. It's not being made into a duplicate of any other strong offensive types, it's just allowing offensive Poison-types to actually realise their potential.
Suppose that idea were attempted. In implementation, it would end up in one of two scenarios: poison would become imbalanced, or the tension between its two roles would end up rendering each less effective. Not every type can be good against a lot of other types. I haven't seen a compelling reason why poison, the poster child of bleeding, would be better taking up a slot in a different archetype.
I don't see why Poison should be restricted to two strengths when you yourself agree that, strategically speaking, the Grass strength is practically meaningless. In fact, strengths against Grass, Water, Fairy & Bug would be more like 2¾ strengths, since Bug is like a slightly better version of Grass anyway. If Bug did become weak to Poison again, though, I'd recommend giving Bug some more resistances to defensively buff it (e.g. to Psychic & Dark, which its already good against), and reinstating its own strength against Poison too.
Making poison good against grass and bug serves more to hurt the power level of grass and bug than to benefit poison's power level. The changes to bug you propose, in addition to being less parsimonious, would also end up just making psychic and dark worse, when they already don't have a lot of offensive firepower.
Poison would be able to serve a much better offense role hitting water and fairy, both of which can be common and perhaps have few other exploitable weaknesses, losing its less useful effectiveness against grass (and bug also being unneeded). Fighting and ground would be nerfed a bit, which in turn would help to bring other types into prominence (steel would become more viable against ice, bug would become more viable against grass and dark, fire becomes a bit better). Flying, by being more distinguished in its effectiveness array from fire, would become more offensively viable. Grass would be less of a butt monkey. Ice and rock would gain defensive niches.
No, I'm sorry, this won't work; half of your choices are completely counter-intuitive. To elaborate, I've included your list below w/comment in red:
Poison: Loses effectiveness against grass. Gains effectiveness against fairy and water. No to losing Grass strength, for reasons stated in previous posts.
Fighting: Loses effectiveness against ice. No; Fighting breaks ice.
Ground: Loses effectiveness against steel. No; soil erodes metal.
Rock: Gains resistance to rock and electric. Agreed.
Flying: Loses effectiveness against grass. Gains effectiveness against ground. Agreed with the first part, but how is Flying's SE on Ground to be explained?
Steel: Gains effectiveness against fairy. Gains weakness to electric (and/or water). Yes to Fairy strength, no to Electric weakness (faraday cage), and maybe to Water strength.
Grass: Gains effectiveness against fairy. Loses ineffectiveness against steel and poison. No to Fairy strength, if fairies = nature (that'd be like Grass being good on itself), and no to neutral on Steel and Poison; Steel is too hard to take much damage from plants, whilst Poison is the antithesis of Grass (hence resistance)
Ice: Gains immunity to ice and resistance to water. Could see water resistance happening (Ice freezes Water, weakening its attack?), but why should Ice be immune to itself?
Also, I don't think Fighting & Ground have problems as big as the others on that list.
I'm not sure what "counter-intuitive" is supposed to mean in any substantive way. It's possible to justify intuitive relations between any types in the chart one way or the other. It's possible to argue justifications for and against any such relation. So your objections in each case are more or less non-arguments.