I do know? I’m a landlord and have had tenants use this option of open banking sharing (it’s not required).
The problem is that as a prospective tenant you do not know and can’t be sure with what happens with your data once it leaves your bank. The particular referencing company you’re using may not give you (the landlord) access, but another one may do, or maybe some other landlords have some special deals with this company that do give them access, etc.
The referencing company’s algorithm may also be wrong or break due to a bug and wrongly reject a tenant (without you or the tenant even knowing which transaction is to blame and thus not being given the option to explain it), especially if you take into account the terrible data quality you get out of legacy banks’ open banking feeds.
There’s also no guarantee as to what happens with the data once your and the tenant’s involvement is done with it. That data may very well be used for other purposes (which benefits neither you nor the tenant). The ICO has clearly set the example that it’s OK to be misusing personal data and the worst that will happen if things come to light (which are a big if given how hard is it to actually report anything to them) is a slap on the wrist, and the FCA will not get involved unless large-scale monetary losses happen.
In contrast, good old bank statements will sit in physical format or digitally in someone’s email, which I’ll trust way more regarding misuse than having it all in a nicely normalized DB with hungry data scientists looking forwards to their next promotion.