I don't think the really cheap scopes bring much to the table where it comes to logic analysis due to limitations in triggering, display abilities and memory. Same for low cost USB 'logic analysers' which offer very little in terms of triggering. If you want to dig deep into a protocol, then a real logic analyser like Tektronix TLA700 / TLA7000 series is a much better option due to having elaborate trigger sequencing. Recently some information got released for these series of logic analysers (
https://groups.io/g/logicanalyzers/topic/109776434#msg803 ) which seemingly allows to create all kinds of wild protocol decoding algorithms to run on the logic analyser itself.
But that is not where it stops. On the TLA700 you can use the TLA7AA4 module which has 125ps time resolution (albeit in a short buffer). On the TLA7012 mainframe, you can use the TLA7BB4 module which has (IIRC) 25ps time resolution (albeit in a short buffer). On top of that these modules have analog outputs which allow to pick up any signal from the probe and feed that into an oscilloscope. Using the trigger output, this allows to show an event both in the analog and digital domain with extreme resolution (assuming using a low noise 12 bit oscilloscope). And I'm not done yet... The modules can use active FET probe heads which have a bandwidth of around 2GHz and which can be fitted straight onto a circuit board using a specific footprint. So if you want to analyse a TPM module, create an interposer board for it which can connect to the logic analyser probe and you can look at the signals through an almost perfect probing system.
Last but not least: these logic analyser systems aren't even expensive. Lowball some of the sellers on Ebay and I'm sure one will bite.