I have a DD who is almost 10. I really couldn't do a playdate/sleepover as a regular thing ever week.
First, I'd feel like I had to reciprocate - I know you say it's not necessary but I would feel like it was and sleepovers are the work of the very devil and I bloody hate them
Second, I work - she's at her Dad's every other weekend, I don't get a lot of time with her, I wouldn't want to give that up for a friend every week
Third, we do a lot of after school activities and a family quiet night in with a take away and a DVD is nice for us to bond as a unit, which I feel is very important because I'm a single parent and I want to be sure that the kids feel that we are a unit.
Fourth, if it had been one of the older kids, I just couldn't have run from pillar to post to take one/drop off - and again, I know you said you do it but I would have felt obligated, or I would have been under pressure to be in the house at a set time for you to drop off and that just wouldn't have worked - when you have an older child at training on a Saturday, for example, and the set up is they ring when they're finished, and that finishing time is weather dependent, having even a commit to be in the house at say 1pm for a drop off would be an issue.
Once a week with the same child for a sleepover I would feel is a bit too much of a good thing - I'd be trying to encourage my child to see their other friends too and if you're inviting him every week for one night a week it's a big chunk of weekend time.
Yes, she should have replied, but she't not mean - I think you're coming across as a bit needy and a bit pushy and I would have backed off if I'd been the other mum.
Can't you find an activity your DS likes to do and take him to that?
Also, sorry, but just because you and the other mum have kids the same age doesn't mean you would be friends - you say you're isolated as the mum of an only - why? You should get activities of your own and develop your own circle of friends and not look to your DS to provide them via his friends iyswim?