Unable to boot my MacBook Pro with message "Need replacement"

It’s time to replace the battery.

Get your Mac repaired and see how much will it cost: Mac Repair & Service - Apple Support 

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My post is not intended to detract from the good advice already posted.

That symbol at Startup, by itself, is not an indication of battery failure.

That symbol is called the Prohibitory symbol. It indicates that major components required to continue booting MacOS are missing, corrupted, or wrong version.

You MIGHT be able to connect the power adapter for a few hours and try again. That could allow you to complete a backup before submitting your Mac for service.

the user diagnostic is a quick test fro GROSS errors. it is not comprehensive. "no faults found" is far from "all is well".

The kind of corruption that could preclude booting can be detected and often repaired in Disk Utility, which is available in Recovery mode.

if you need guidance, readers are eager to provide it.

NOW if you re-boot, and get the Prohibitory sign, you should use Recovery to re-install MacOS.

By design, a re-install in place does not deliberately change any User files or added Applications. But it does bulk-erase System directories and re-write about 350,000 files.

NOTE: the Volume MacOS Base System is NOT a place you can install. It is a compressed disk image pulled from the Mac ROM to run the Recovery software, including Disk Utility and Installer.

Sources suggest that model is only supported directly by Apple to run up to 10.13 High Sierra. It can't run Catalina DIRECTLY.

So you will likely have to install an older version normally, then upgrade to High Sierra. If you choose to hack it to an unsupported version, that's your choice, but will likely have to be done later.

Since you can never install older over newer, this will likely involve an ERASE step. You will need a backup.

¿What is the date of your most recent local disk-based backup, and by what method?

The OS is recognized as bootable, but is not compatible with that Mac for some reason (wrong OS, or the OS is damaged). For a 2011 Mac, a failing Hard Drive is a very likely possibility.

You can try running the Apple Diagnostics to see if any hardware issues are detected other than the worn out battery. Unfortunately the diagnostics rarely detect issues especially with the drive. There are two other options for checking the health of the Hard Drive. One involves booting from an external full macOS USB drive, and the second involves creating & booting a Knoppix Linux USB stick. Both options allow you to use a special utility to check the health of the internal Hard Drive.

Hopefully you have a good backup & don't need to recover any data from the internal drive since the more you try to fix it or access it the more likely the failure will get worse if the Hard Drive is indeed failing. If you don't need data from that internal drive, then you can try what you like. You can try using Disk Utility First Aid while booted into Recovery Mode (Internet Recovery Mode with Command + Option + R will probably be needed) and/or reinstalling macOS over top of itself. This is the longer way to deciding the drive is bad.

Of course there could be some other issue, but a failing drive is most likely given the symptoms.

The Battery issue is not contributing to the problem since the laptop is powering on & trying to boot.

The older Samsung EVO SSDs have not had a great track record of compatibility when installed internally in a Mac. It could just be a compatibility issue being exposed.

If you still have issues after reinstalling macOS over top of itself, then I would highly recommend checking the health of that Samsung SSD as I mentioned in my previous post. It may provide some clues.

You seem to be booted into a very old macOS installer. What version of macOS is actually installed on this SSD and what version of macOS will that installer actually install?