Figure it yourself.
1) The speaker does not need to be grounded to work. Any more than a flashlight or pocket radio.
2) User safety demands the secondary be grounded. What if the primary B+ insulation breaks down, and someone is touching both speaker and a lightswitch or radiator? Grounding the secondary ensures no shock (altho maybe fire).
3) Many negative feedback schemes expect the speaker side to be referenced to a common point, "ground" for slang. (Your amp has no NFB from the speaker.)
99% of tube table radios had grounded speaker for shock-safety; it was a check-point in UL tests. Sometimes built-in: one side of OT or speaker not even brought out except through a metal mounting foot.
The vast majority of tube Hi-Fi had grounded speaker, for safety and also for NFB.
Some "Professional" tube PA amplifiers had ungrounded secondaries. The system designer was expected to design the safety and signal grounding to suit requirement, and document a strapping scheme for the installer. Speakers on top of poles at a sport stadium might not be a safety hazard, but are subject to lighting strikes, which changes the grounding goal.
You are not doing anything clever/goofy. Ground one side. Yes, on paper going OT to ground and ground to speaker is all the same thing. In practice, especially on a rusty chassis found behind the surf-shop, you want to ask if you can trust those connections, both for safety and for not crapping-out on tour.