It might be a case that you get what you pay for, the budget is a tad low.
Cheap test equipment could be a "Highway to Hell" with misleading results, but good test equipment is more like a "Stairway to Heaven". Those two songs might also say something about the ratios of the numbers of people going each way.
In any case, in my opinion, the best ESR meter in the world was designed by Bob Parker. These came out as a kit in Australia years ago, they are about $100 AU dollars, , roughly 75 USD, but you would have shipping:
https://www.altronics.com.au/p/k2574-esr-meter-kit/
They are also available on Ebay and go under the name "Anatek Blue", in assembled or kit form, but I really would recommend the Altronics kit. It has a sturdy case with a painted and detailed screen printed Steel panel, the printed ESR data for the capacitors is actually more helpful and quicker to use than a graph, and Red LED's with a red acrylic cover, very nicely done for a kit. The Anatek ones never struck me as being as robust.
The reason these are so good is that they use a micro current pulse to measure the capacitor and the voltage developed across it is very low. They give very reliable results with in circuit testing, especially in computer & other SMPS units . Also, they have a lovely LED display. In addition, I have other ESR meters. One type uses a sine wave to perform the analysis, it is not nearly as good, Dwight mentioned this issue with the sine wave frequencies, the pulse design of the Bob Parker meter avoids this problem.
Plus the Bob Parker meter is very accurate at measuring low Ohmic resistances, that a standard DVM cannot do, and which logically you would expect a high quality ESR meter should be able to do well.