Managing Large Shared Mailboxes in Exchange Online – Performance Strategies and Trade-offs

OWA is the easiest solution, assuming all the features you need are supported therein. Use the Open another mailbox functionality (assuming Full access has been granted) or just use direct links to open the shared mailbox (i.e. https://outlook.office.com/mail/email address removed for privacy reasons/). The new Outlook still has to add proper support for some shared mailbox scenarios, but it is also an option to consider.

Second to that is reducing the mailbox size, either by cleaning up older items or enabling archiving (subject to license availability). Configuring the shared mailbox as account in Outlook can also help in this regard, as it will expose the cache slider, and even more granular sync controls on the folder level. You do not need a license for that btw, just full access permissions.

Whether DLs or Microsoft 365 Groups are an appropriate solution I cannot tell, you have to decide on that based on the functionalities you need and the use case scenarios.