JA heroines often go out in public with unmarried men. It was spending too much time with a particular gentleman that was frowned upon. I do not believe anyone worries Elizabeth will be attracted to lower class men. Elizabeth, IMO, would not be interested in any man that was undesirable due to lack of rank or fortune. I am doubtful she would want to even meet lower class men or that Mrs. Clay would suggest an introduction to one. Elizabeth is a great snob after all; think of her cold reaction to Captain Wentworth marrying Anne. In Chapter 1 Elizabeth “would have rejoiced to be certain of being properly solicited by baronet-blood within the next twelvemonth or two.” Elizabeth has narrowed her choice down significantly to baronets; she wanted from childhood, as did Sir Walter for her, to marry his heir Mr. Elliot.

Sir Walter did encourage Mr. Elliot towards Elizabeth years before but the man was completely uninterested. Sir Walter thinks Anne too faded and thin to ever marry and he believes Elizabeth’s beauty assures her an appropriate marriage at some point or another. I think you are right that Sir Walter enjoys Elizabeth running his house, from Chapter 1, “Elizabeth…being very handsome, and very like himself, her influence had always been great, and they had gone on together most happily.” I don’t think Sir Walter feels responsible to do more for his daughters on the subject of finding husbands—it is probably selfishness and laziness combined. :)