Is there any evidence the FBI pursue any sort of investigation aimed at trying to determine the existence of these tapes and texts?

It's a serious question. If they seriously investigated and found there simply is nothing supportive, then lets hear a few details about said investigation. If they remain silent then perhaps this is another case of not wanting to pursue leads that might prove embarrassing to the Bidens.

What would s serious FBI investigation of that look like to you?

The MO is to manufacture one unverifiable accusation after another to tantalize and enthrall the base that corruption exists that isn't being investigated and it has to be doozies nowadays because fearless leader is being closed in on. You want a serious investigation of hot air, they'll look and say it is hot air, but by that time another unverifiable accusation will be manufactured, or it will be implied someone "disappeared", and... off to the next, with this one revived and intertwined with the new. After a while you turn a deaf ear to the din and tell yourself my time is better used not following/attempting to understand vapor. Always willing to look that direction if something solid turns up, but you have other things better to do.