Dunno if you own an AMD product, but since last year, AMD implemented a lot the Relive recording/streaming software suite.
AMD users already have Raptr, GeForce Experience features and far beyond include into the drivers now.
If i remember well, the Adrenalin driver brought these features came in September last year.
You can check AMD youtube channel, there are a lot of videos explaining how to use the Relive recording/streaming suite and the Radeon Overlay.
At the moment the AMD software is pretty good and pack the same things that the GeForce Experience, Raptr and even beyond.
You do not need to log in to use the Relive recoding/streaming suite, it is packed into the Radeon Settings Panel easily accessible.
You can tune and optimize your games profiles and manage your gpu clocks as always directly from the Radeon Setting Panel.
With the Radeon overlay you can easily manage your recording/streaming session while gaming.
Finally content sharing through different social platforms is very well implemented into Relive suite.
So if you purely look at the driver software package, the AMD Adrenalin Driver suite is really pretty good and offer to users a wide range of features for their AMD gpu's that even the GeForce Experience can't deliver.
If your question was regarding purely the drivers games performances and per game performances optimisation, i think the AMD drivers are on par with the Nvidia one regarding follow up and optimisations for new games.
I have a HD 6670 GPU so I can't use the Relive or Adrenalin driver, AMD should make a software like Raptr that all AMD users can use
AMD Athlon II X2 250 3.0GHz
8GB DDR3 RAM
MS-7597 ver 1.3
Sapphire Radeon HD6670 1GB GDDR5
Windows 10 Pro 64bit
1600x900 screen resolution
100/6 internet speed
there there WERE drivers that shipped with Raptr - called "AMD Gaming Evolved"
Unfortunately you are right, the HD6000 series do not support the last AMD drivers and relay on old legacy drivers.
Nevertheless i would advice you to sell your HD6670 and look for a second hand cheap HD7750.
Dunno if you can afford to swap gpu, but any gpu past the HD7750 included, will support the Adrenalin drivers and so Relive.
Again i understand you, but your card is fairly old now even it still works fine, it is hard to bring support to all old cards.
I would advice you to try to upgrade gpu anyway to a HD7700 series and/or above, that support the latest drivers and will bring you huge performance boost over the HD6670.
Obviously if gpu upgrade is possible to you and you can find easily second hand gpu's.