I concur with Richards thoughts on this...and he in fact knows much much more about the chassis types than I ever did because when I was employed in the machinery world, 20+years ago, we used to see many B&S engines but actually not so many on Merry Tiller.
Richard has Merry Tiller as one of his specialist subjects.
I haven't visited B&S site for a month or two...but had noticed then that the once easy location of their massive range of scanned engine parts books and operating manuals was less easy to find.?
The engine date is all in the Code numbers.
B&S recommended in more recent times Champion J19LM as the spark plug to use, gapped to 30 thou of an inch.
Choke slides on some MT B&S engines seem to wear loose and thus it becomes difficult to hold them on absolute full choke for cold starting.
Air filter may be worth checking, a worn/blocked filter can have caused a previous owner to try to adjust carb to correct over rich running.
A weak ignition can sometimes affect starting, especially at a full thottle setting, there being more load on ignition at full speed. A weak ignition engine though may fire once only and not be able to sustain a run...whilst you have achieved a run with yours hence the carb/pump problem thoughts. Weak ignition can sometimes also cause a false rich fuel mixture effect leading to an incorrect carb' weak adjustment.
I would also tally with the thoughts on consulting a long established local machinery agent if you feel unfamiliar with working on the machine. Given that your purchase sounds in good basic shape it may be well worth getting it up to specification professionally.
Apologies for my absence.
Life and work just gets busier...
...and now I must away and mend next doors washing line.
...due to an errant move with my hedgecutter whilst trimming of our Beech hedge.