power supply has likely died, new one is easy to get.

https://www.bestbuy.com/site/evga-br-series-700w-atx12v-eps12v-80-plus-power-supply-dc-dc-technology-black/6346160.p?skuId=6346160

A couple of PSU replacement videos:

https://m.youtube.com/watch?v=d5c0FRevVb0  790 is 7010's predecessor.  Same connections.

https://m.youtube.com/watch?v=jUbOy9r8BAc 

Connections are straightforward.  24-pin PSU connector goes to 24-pin connection on MB.  Hard to screw that up.   Connections for your optical and HDD/SSD drives will be easily recognizeable.

If you do decide to go with another 7010, you won't have any trouble transferring your drives.  If you have Win10, chances are it will detect new hardware and ask you to reactivate online.  Shouldn't need a product key.  The same for MS Office if you have it.

Optiplex 7010 Tech Guide 

Dell Optiplex 7010 Manuals Page 

your 7010 MT top cpu is i7-3770 (socket LGA 1155).  chipset is Q77.

similar Dell mini tower models for this cpu include XPS 8500, Inspiron 660.

XPS 8500 is H77 chipset, Inspiron 660 is B75 chipset.

you can google small different between these 7 series chipset.

XPS is slightly higher tier than optiplex, whereas inspiron is consumer mainstream.

optiplex is business class.

Per your 1st post, it sounds like the PSU is on it's way out.  Been there, done that.  By the way, power light blinking orange (not flashing an error code) is typically the PSU in an Optiplex.

Thanks. I'll look for the XPS and Inspiron models as well. How about the OptiPlex 3020 and 9020 series? My 7010 CUP is Core i3-3240 (3.4GHz, 6MB cache), and is fine for my needs.

I am looking for a model that I can easily install the upgrades I have on my 7010: SSD drive, additional HDD, etc. IOW, I would like to use the existing caddies, brackets, etc. without having to search for new ones.

Thanks for the vids. I'm also concerned that the problem may not be the power supply. Hence the hesitation to replace it.

The best Optiplex from that generation is the 9020 with i7-4790.  Same chassis; you can use your parts.

Besides what I mentioned before with transferring drives:  Use Dell SupportAssist  to download any needed drivers.  Bare minimum you'll need chipset and graphics drivers.  Screen might a little funny looking, but usable until graphics driver is downloaded.  Also, you may find more convenient:  https://www.dell.com/support/kbdoc/en-us/000122023/optiplex-9020-windows-10-driver-pack 

Optiplex 9020 Tech Guide 

Correction - Tops is the sporty XE2 from that gen.  Can Google:  Optiplex XE2 Tech Guide and Optiplex XE2 drivers cab file.  Cab file will lead you to Dell site.  XE2 manuals can also be Googled.

Re: My 7010 CUP is Core i3-3240 (3.4GHz, 6MB cache), and is fine for my needs.

that is an entry level cpu, 2 core.  since that is good enough for you there is really no need to upgrade to newer gen intel cpu.

replacing MT psu is not technically difficult.  there are 4 standard screws fastening the psu to back wall of chassis.  24 pin and 4 pin are the basic connectors to motherboard and cpu.  additional power connectors for sata ssd/hdd.

More accurately, "...cab file" will lead you to Dell site.

I just ran the BIST test. Pushed the BIST button, the LED lit up and the fan came on. Test passed. 

https://www.dell.com/support/kbdoc/en-ca/000125186/how-to-troubleshoot-power-issue-on-a-dell-desktop-or-all-in-one

Since the problem has not resurfaced, should I still replace the PSU? There are many cheap used Dell PSUs for 7010 on eBay. Maybe just replace with one and see?

What that means is your other internals, and anything connected externally passes the test.  Smart thinking doing the BIST test.

I had to edit the above pgh. because my phone keyboard sometimes changes words on me.

A bad PSU, or a PSU going bad, can still whirl up the fans and give a green light.  It can be a little deceiving when a PSU is the problem.

Going with another 7010 might be the best to fit your needs.  Mine smoothly runs recording studio software.  It came with 8GB and I made it 12 GB.  I also had a PC that didn't run the software smoothly until modifying it.

In the screengrabbed charts below, it shows tbe year certain models, including Optillex's, were introduced.  One example, my 7010 is from 2014, but was introduced in 2012.  The charts might also help you make a decision.

If you are satisfied with your 7010, and you want to run it until problem resurfaces, nothing wrong with that either.  With chances are it being the PSU, problems will likely resurface.  I had a PSU do the same thing.  Do involuntary restarts while using, run good for a while or short while, then act up again and eventually die.  Yah, PSU's have no user serviceable parts inside.  Dangerous if one opens it up and doesn't know what they're doing.

Power supplies don't die Right away always.

This unit is showing signs of impending death.