iMac super slow to start up - would a external SSD help?

the only thing I saw in your etrecheck report that would explain the slowness is the very slow (5400 RPM) HDD. using an external SSD would definitely speed things up. additionally, it would be well worth your investment.

if needed, please see Use an external SSD as your startup disk with your iMac or Mac mini - User Tip to set it up.

Besides the slow internal hard drive you have a few other issues that need attention.

1) Time Machine has not been updated in a very long time.

2) iCloud sync is hung up and pending.

3) MS Apps are crashing.

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If you want to manage the iMac for you daughter? You should fix the above issues or erase and reinstall.

see > Use Disk Utility to erase an Intel-based Mac - Apple Support

and > How to reinstall macOS - Apple Support

If you want to give the iMac to your daughter? Then you should clear it off and setup her as the owner.

see > What to do before you sell, give away, trade in, or recycle your Mac - Apple Support

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To answer your internal HDD versus external SSD startup question, the answers is YES it will startup and run faster.

see > Use an external SSD as your startup disk … - Apple Community

Just keep in mind that starting up and running from an external USB SSD, may not be the best setup for young, unskilled or haphazard users.

Yes, the external boot SSD option is definitely indicated. Your current mech hard drive is doing this:

Performance:

System Load: 1.54 (1 min ago) 1.64 (5 min ago) 1.75 (15 min ago)

Nominal I/O usage: 0.57 MB/s

File system: 48.68 seconds

Write speed: 63 MB/s

Read speed: 43 MB/s

The speeds for the least expensive USB3/SSD option will be 400MB/sec. It will feel like a newer computer.

The external SD makes sense in this case because when you move to a newer computer, the external can be used for extra storage or backups on the new one. Retained value.

That is the only pay-for option that passes the financial sniff test in your case. Your iMac was never going to be fast anyway. It is the crippled educational/institutional model with downgraded hardware to be attractive to cost-conscious bulk purchasers.

In addition to the slow mech hard drive, it has a laptop-class 2-core processor running at 2.3 gHz. The base consumer model the year (US$200 more) came with a desktop-class 4-core processor running at a minimum of 3.0gHz. These benchmarks from the MacTracker database give a scale of the performance reduction:

This forum article describes photanalysisd and how to manage its mega-maniacal tendencies:

https://apple.stackexchange.com/questions/310594/what-is-photoanalysisd-and-why-is-it-using-77-of-my-cpu