Compact Freezer Repair: Thermal Overload Protector Replacement

[Tea Sea]

Haier HF50CW20W 5.1 cu ft freezer https://www.haierappliances.com/appliance/5-1-Cu-Ft-Capacity-Chest-Freezer-HF50CW20W stopped cooling. I replaced the starter relay with Bunn 39912.0002 https://www.amazon.com/dp/B07572HJYS?psc=1&ref=ppx_yo2ov_dt_b_product_details , which appeared to be an exact match, because the consensus on YouTube was that was the most likely fix, but it didn't fix it.

Now I'd like to replace the thermal overload protector, but searching the number stamped on the one with it (T0377/05 C531) turns up nothing. The compressor is ZEL HDL140D 115v 60Hz. According to this document https://frigopartners.com/media/04/cf/ec/1636646991/catalog_zel.pdf (page 3 in the 115v/60hz North American Range section), the compressor is 1/8 hp. The closest thing I can find online is this https://www.ebay.com/itm/133401689447?var=433190705658 which is 1/8 hp and 115 volts, like the ratings on the compressor. I have read that the capacitance and ohms need to be the same, as well, but have no idea how to find that information, or really what it even means.

Is this part a suitable replacement for the overload protector? Is there any potential danger or harm if it is not? I know little about these things. Thanks in advance!

Background information: doing this for an elderly relative, who said it stopped cooling. I went there and plugged it back in, the compressor came on, and it froze a pan of water for 24 hours, but then stopped working again. Now it does nothing when I plug it in, no clicks, no hums. Since I have not been with the unit, that's about all I can offer.