Im new to this forum but reading this triggered my response as reading your post reminds me of me and my girlfriend / when she was trying to really understand and help me before we bought a house etc.
You say your bf is honest and tells you everything. Trust me when I say I am as direct & cut throat as anyone.. and I would tell my GF barely 10% of what truely went on. He gives you his wallet etc when you are out as he would rather gamble without somebody on his shoulder & tries to keep his addiction away from you as much as he can.
It sounds like you have no true idea about his finances as he tells you its his business etc. My advice is if you want a house & future together, you need a joint savings account that you can put X amount away in each month each which he cant access. The only reason I managed to afford a house is because I could never access & bet with the joint account!
Goodluck! You are young and my girlfriend would definitely tell you its never easy dating an addicted gambler (which you need to realise he will forever and always be addicted to). Its one thing when you are living separate lives, but wait until his messups are then your messups. If hes not willing to give you more control / coming up with excuses, you have a long rd ahead!