Safari suddenly slow on Sequoia

First the important part of backing up your data. Yes I would recommend copying the contents of your Documents and Download folder to an external hard drive. Not sure if you use iCloud for photos or files, but those will remain safe, although it is always a good idea to have a backup of those as well. I do not recommend restoring from a Time Machine backup. Generally there is no problem using using Time Machine as a backup and I do it too, but the purpose is really to restore the computer to the exact same state as before the reset and that is not want you want to do in this case.

For the Reset, follow the steps here. Continue to use the same Apple Account as there is no harm there:

Erase your Mac and reset it to factory settings - Apple Support

For the legitimate apps that you do use such as the Microsoft products, DaVinci, WhatsApp, etc, make sure you download them from either the App Store or the Developers website if an App Store version is not available. Drag the files located in the Documents/Downloads folder you saved to the external hard drive back to the new Downloads/Documents folders. Don't replace the Documents and Downloads folders from those on the external drive, just move the contents.

You do want to use a backup and I would recommend to use Time Machine at this point. Once you have verified your files from the External drive have been moved to your computer, you can then set up the drive for Time Machine. With Time Machine, it uses an entire partition of the drive so when you follow the steps to set it up, it will prompt to erase the drive to continue and it is not recommended to store other files on the drive.

Back up your Mac with Time Machine - Apple Support

Others may suggest a different backup solutions such as Carbon Copy Cloner, Super Duper! and Backblaze and there really is no problem with any of those either and some would say advantages. I do use Time Machine and it has worked quite well. The one caveat I would say is to stay away from CleverFiles, Disk Drill, MacPaw(CleanMyMac) products.

Post the free EtreCheck report using the Additional Text option when posting. The report contains no personal information.

How to use the Add Text Feature When Post… - Apple Community

It is also not clear from your tests if you tried using Safari without signing into Google. The only thing noted was that Safari asked you to sign into Google. And also if you tried a different search engine other than Google. Google has an incentive to prefer their Chrome browser when using their service, both when signed into your account and also to tailor specific search results based on data they serve up from your account. Since the problem is limited to a Search Bar delay, that would be the common denominator that may have been overlooked.

What you are experiencing is not what has been noticed by others on Sequoia, or even third party comparisons of Chrome and Safari, so I suspect there is something unique to your system that is causing the slowness. You have 2 choices, attempt to isolate the problem like you are doing and remove any system conflicts that may show in the EtreCheck report or perform a Factory Reset and compare your results before restoring everything from a backup. Once you find the problem does not exist on a clean install, then start adding things back until the problem appears to identify the problem.

Unfortunately you have some of the most problematic apps installed on your computer. I would expect you are seeing slowness far beyond just what is noticed in the Search Bar of Safari. I suspect many will tell you the best option for you is to Factory Reset the Mac by doing a clean install. You would need to backup any of your data that you want saved to another source.

Just to highlight some of the problems noted:

• System Integrity Protection is Disabled. This allows modifications to system files and opens you up to Malware.
• Secure Boot is reduced. This was probably required by one or more of these files installed to make booting vulnerable and
• /Library/LaunchDaemons/com.redgiant.service.plist is unsigned. It was one of the files that needed to be bypass Gatekeeper in order to be installed. Unsigned files should not be installed on your computer.
• /Library/LaunchDaemons/com.macpaw.CleanMyMac4.Agent.plist is unsigned. Also required Gatekeeper bypass. CleanMyMac is known as BrickMyMac here and you can use the search option on the top of this page to review others that used the software and left their Mac unusable.
• /Library/LaunchDaemons/com.borisfx.rlmserver.plist is unsigned and also required Gatekeeper bypass.
• ~/Library/LaunchAgents/com.macpaw.CleanMyMac4.Updater.plist is unsigned.
• CleanMyMacX should be removed. It is best to first attempt an uninstaller to remove remnant files, but CleanMyMac wants to live on your computer so removal can be difficult. The app resides in the Applications folder. It is also set to run at Login consuming valuable resources.
• System Extension ProtonVPN WireGuard is installed on your computer without the companion app. This means it was not properly uninstalled to remove the Extension. Those can be difficult to remove.
• System Extension SpyHunter Guard installed. Again can be hard to remove without an uninstaller and the app rested in your Applications folder as well.
• Kernel Extension SecureDisk.kext - com.cleverfiles.SecureDisk used with Disk Drill has always been problematic. Just as System Extensions, Kernel Extension is best removed with an uninstaller.
• Kernel Extension WondershareDriver.kext - com.wondershare.driver installed and linked to the Recoverit app in your Applications folder.
• /Library/Application Support/CleverFiles/BackService.app
• Library Extension ufsd_NTFS.kext - com.paragon-software.filesystems.ntfs. Not sure if it is being used anymore and don't have any specific recommendations other than seen this also on systems running poorly.
• /Library/LaunchDaemons/com.cleverfiles.cfbackd.plist
• /Library/LaunchDaemons/com.enigmasoft.spyhunter.shmhelper.plist
• /Library/LaunchDaemons/com.paragon-software.ntfsd.plist linked to the previous Library Extension.
• /Library/LaunchAgents/com.paragon-software.ntfs.notification-agent.plist

I understand this is a long list and not exactly what you are looking for. I am now more convinced than before that the problems you are experiencing are isolated to your computer and not a system problem with Mac OS or Safari. If you frequent third party sites that operate as a "store" to download free software, you should refrain from doing that as many also contain malware which requires bypassing Gatekeeper to install on your computer. I do believe the best course is a clean install as it will be a long process to remove all of these problematic files and the Extensions are the most difficult to remove.