Although BA really devastated the sweet spot of the program (upgrading from WT+ to CW using Avios, plus lowering Avios earning on WT+ fares by 25%) now with drastically raising the number of Avios required - if upgrades are your priority, and you prefer to go through LHR buying premium economy for those ATH flights, it's still best to stay with BA. Get the BA credit card and use the Avios earned with it to offset the increased redemption cost.
As long as you don't buy economy fares, especially on long-haul travel, you will still qualify for Silver.
The alternative would be AAdvantage. While you can upgrade BA WT+ flights with AAdvantage miles, the value proposition is very low: AA requires the BA premium economy ticket to be in "W" fare class in order to be eligible for upgrading. That's full-fare if bought on BA ticket stock, and usually more expensive than a discounted business ticket.

If you switched to AA, it would make more sense to just buy economy on the US flight via PHL and use miles and co-pay to upgrade to business going to ATH. However, that would preclude you from earning Platinum /oneworld sapphire status as you'd be missing out on the extra EQP the WT+ ticket on BA earns. You will not qualify on miles with the amount of flying you stated.