problems after insertion of cheap SSD into older 2008 MBP

Most current SSDs are SATA III devices and some may not properly negotiate the link speed with some older laptops which only have a SATA II controller.

It is possible the internal hard drive cable is defective and needs to be replaced when using an SSD.

Remove the SSD and use an USB to SATA Adapter, dock, enclosure to boot the SSD externally. The USB 2 connection will make the SSD feel slow, but it should be stable for basic work loads and testing. If the SSD works well when booted externally, then one or both of the above could be the issue.

You can try unchecking "Put hard drives to sleep when possible" located in the Energy Saver System Preferences. Some SSDs have problems waking from sleep mode.

Some cheaper SSDs may leave out some other optimizations to lower costs which can greatly affect performance. You need to be careful buying cheap SSDs. Be careful purchasing the cheaper low end models even from reputable name brands. While the Crucial MX500 SSD works well with Macs (at least the 2010+ models), the Crucial BX500 SSD which is an inexpensive basic economy model removes a lot of "features" and the drive can perform extremely slow in some situations where the speed is the same as a traditional hard drive. OWC makes a 3G model which is a SATA II drive so negotiating the link speed is not an issue with it.

If you buy a cheap non-name brand SSD, you run the risk that the quality of the internal components are sub-standard and are more likely prone to failure since these manufacturers are focused on "cheap".

If the SSD supports SMART, then you can use an app such as DriveDX to examine the SSD's SMART attributes to look for issues. If a drive supports SMART, then you may be able to run the drive's internal self tests. You can post the DriveDX report here using the "Additional Text" icon which looks like a piece of paper. Otherwise, there really isn't a good way to test SSDs.

If the SSD supports a hardware secure erase feature, then in addition to erasing the SSD it will reset the SSD to factory defaults which can sometimes "fix" a misbehaving SSD. The tools to activate the hardware secure erase are only available using Linux.

That makes perfect sense. Even some good SSDs can slow down when writing a lot of data to them in a short period of time such as installing an OS or cloning a drive. SSDs include a faster write cache to help improve performance since the actual SSD NAND memory is actually quite slow. Once this write cache fills up, then you experience the speed of the actual slower speed of the SSD. With some SSDs, performance can drop to that of a traditional hard drive. It can take a while for some SSDs to recover and resume the "faster" speeds which utilize the SSD's write cache. This can also affect read speeds during this period as well.

The cheaper SSDs have a smaller and simpler write cache that can be exhausted much more quickly. The better SSDs will still have a drop in performance usually dropping to about half speed, but the better SSDs recover much more quickly. How much of a drop can vary from model to model and manufacturer to manufacturer. The write performance of many SSDs will drop after about a minute of sustained non-stop writes which would include GBs of data.

On cheaper SSDs, you may see this happen again if you download or install any extremely large files or apps (GBs in size).

If you continue to have problems, then post a DriveDX report here so we can evaluate the health of the SSD. DriveDX also allows you to run the SSD's internal self test, although the extended (or long) selftest is not available in the trial period.

You should probably enable TRIM support on the SSD to help with performance.

https://www.maketecheasier.com/enable-trim-for-3rd-party-ssds-mac/