If I understand it correctly, you decided against a newer model and got a new battery for your 2011 MBP?
Since I use a similar older MBP 17' from 2010, I am very familiar with that problem and how to optimize it:
Here is a short wrap up of what worked for me so far:
#1 Make sure your cooling fan has no dust built up inside it's cage and most important no dust build up inside the little heat pipe fin cooler that goes infront of your cooling fan. This is very important. If those fins on the heat pipe are clogged, your cooling doesn't work and your device heats up.
#2 Since you open up your MBP anyways, make sure your CPU/GPU thermal compound is renewed. Most compounds dried out over the years of usage, so you want to renew thermal compound. This will shave off 10° or more if your compound has never been renewed. All this reduces heat load which is important with newer MacOS installs.
#3 Check on the latest OCLP version. This is very important, because the software engineers may figured out how to add GPU support in their latest version. This happened to me and I was very very lucky with the last version of OCLP. It brought back GPU support for my Nvidia M330 card from 2010. It made a huge difference. My latest macOS Monterey works like a charm again. However this depends on your GPU or hardware configuration. If you do not have an additional dedicated GPU, still check out those new versions for non dedicated GPU Chip support.
#4 Tune Up/Optional:
I always try to keep my system in a workable range when it comes to RAM Memory pressure. Unfortunately I can not upgrade my MBP to more than 8GB Ram. I do always recommend to put the activity monitor on the Ram setting to see if you have any memory swaps. Swaps should be avoided, so close windows and programs to go back to a "No-memory Swap" setting, which is visible on activity monitor under RAM / Memory section.
If you don't have Swaps, your machine will run a lot better.
#5 A cooling mod that has worked for me very well is this method:
On my 2010 17' it worked out real great and cooling fans do not come up as often in the past.
During work the laptop gets how of course, because now your bottom case acts as a heatsink. Since your battery is encased in a plastic casing on your 2011, there is nothing to worry that the no hotter bottom plate heats up your battery. That's also the reason that this cooling mod should be avoided if you have newer MBP with battery that are no longer encased in a plastic box like our older models still have.
And since your bottom case is getting hotter now, finally a laptop cooling stand will work because it blows onto the bottom case.
If you keep those options in mind and combine a real new OCLP install with no-swapp & cooling mod, you have a good starting point from there to further optimize things.
Hope this helps, Greetings from Germany