So much sub-text going on that I don't understand.
What subtext?
It's as plain as a pikestaff, how could you possibly be confused by women's fear of what men are capable of?
Over 90% of women & girls experience sexual harassment or assault.
That makes us wary.
YOU may be a thoroughly decent chap. But WE don't know that.
That makes us wary.
The vast majority of rapes are perpetrated by men we already know, not total strangers
That makes us wary.
You know the Schrodinger's Cat thought experiment, right?
geekfeminism.fandom.com/wiki/Schr%C3%B6dinger%27s_Rapist
That's not to level accusations at individual men btw, so please don't feel 'got at'.
If you genuinely want to understand -
researchtobedone.wordpress.com/2012/10/18/for-those-who-dont-understand-schrodingers-rapist/
It's also not just the fear that possibly someone we don't know well could be a rapist. It's that he might be sexually entitled, or coercive, or a low-grade sex pest, or a sulker, or a sleazey commentor, or only interested in the content of our pants instead of our marvellous personalities 
And that makes us wary, because it's downright unpleasant, fending off a man who won't hear our first "no".
It's exhausting, navigating this. And it's at the back of our minds, ALL OF THE TIME.
I am sorry about your bereavement.
You must have had a really hard 2 years. 
So when you feel ready to start dating again - bear all the above in your own mind. Also this -
We feel safer around men who let US set the pace. Who are genuinely interested in women, not just women's bodies. In women as a set of people, not just one individual you happen to fancy. Who read women's literature, watch women-led & directed films, are aware of women's political issues, women in history, women in science, sport, or industry.
But I feel that having someone visit my home or me visit theirs, gives a pretty good indication of personality and so is quite an 'open' thing to do. Should I reconsider?
Nothing shouts "I expect a shag, & that might be the only thing I'm after" than an early invitation to visit a date's home.
Of course you can ask, after a few dates, but the onus is on you to prove you are trustworthy, not on her to instantly trust you.
Some women are bolder than this, but for many of us ... once bitten, twice shy.