Does changing Iphone battery improve performance?

Protect & Prolong life of your iPhone's Battery:

1. About the battery usage on your iPhone, iPad, and iPod touch ...
2. Batteries - Maximising Performance - Apple (IN)
3. iPhone Battery and Performance

Try some of these options too.

1. Avoid Screen Mirroring
2. Put on Low Power Mode
3. Put Auto Lock On (1 min):- with low power mode it will be 30 Sec
4. Put Auto Answer Off
5. Disable Push notifications for apps you do not use often
6. Keep your iPhone Face down to prevent notifications to light up screen
7. Enable Auto Brightness:- Settings > General > Accessibility > Display Accommodations > Auto-Brightness [Enable]
8. Manually lower brightness from Control Center
9. If you Want to watch continuous Video. Stream it on TV or a Monitor
10. Don't force close apps; Apps which are not on frontend screen are paused by the system to save energy. Apps require more energy to re-launch. Apps which are open in the background use large memory is a myth and quitting them is counterproductive
11. Location services to be allowed only while using the app
12. Set your iPhone to fetch email and calendar data from internet less frequently
13. Turning off Background App Refresh 
14. Charge before it drops below 20-30%
15. Try connecting to trusted Wifi where ever available in stead of Cellular
16. Enable Airplane mode in areas with bad cellular and wifi coverage. Your phone keeps trying for reconnection with the missing / weak network. This uses lot of energy
17. Switch off Wifi & Bluetooth while in Roaming or Not in Range
18. Enable "Wifi Assist" : Settings > Cellular or Settings > Mobile Data > Wi-Fi Assist [Enable]
19. Periodically Check your Battery Health (Settings > Battery > Battery Health)