would it be worth obtaining a tail wheel conversion
Yes. As you say, it's an opportunity to try something new while accruing hours.
Re the Tiger Moth
A wonderful aircraft to have flown. (Note the past tense.)
However, the reality is that although Brits tend to wax lyrical about the Tiger Moth, the Belgian Stampe and the American Stearman, which were designed and built in the same era, are better aircraft.
I was thrilled when I first flew a Tiger Moth at the Tiger Club, and tootling over the countryside in a classic open-cockpit biplane is an enjoyable experience but, after I'd been checked out to fly the Stampe, only took a Tiger Moth if there wasn't a Stampe available. It's much more responsive (ailerons on both wing surfaces) and much more fun. Perhaps if I'd done more hours in the Tiger Moth I'd have a different view. It's relatively easy to fly, but difficult to fly well - I didn't get to that stage.
The Stearman is a wonderful aircraft, a joy to fly. Significantly bigger than the other two but excellent handling qualities.
In order of preference of that vintage, I'd say Stearman, Stampe (almost a tie, but for different reasons) and then Tiger Moth.
Chipmunk
I agree with Genghis. I learnt to fly on Chipmunks and, like everyone who did, consider myself very fortunate to have had that opportunity. It really is a wonderful aircraft - perfectly balanced.
Similarly also considering aerobatics
Well worth considering.
Even if you don't progress to serious aeros, aerobatic experience is very useful. It's a great confidence builder because you get to know the aircraft and get used to unusual attitudes - and recovering from them when a manoeuvre doesn't work out as planned.
as well as other war birds
When you've built up some experience in one or more of the above - particularly in a Chipmunk - I recommend flying a Harvard. I used to have a third share in this one.
I had serious misgivings when we sold it and have regretted doing so ever since.
FL