This is a great question. Let me take it in parts.
First off, you're right that the most important place for support is via companies. Ultimately, that's what matters. But you'd be surprised how much voice people have in influencing companies. For example, if you're an employee, I can tell you first hand how closely CEO's and executive teams monitor (for example) what gets asked about at all-hands meetings.
Shareholders matter too. Companies want to know what their investors think and many of them have whole "investor relations" departments whose job is, in part, to relay investor sentiment to the rest of the company.
In terms of trading, yes, you can specify what exchange your shares trade on. Almost every major broker is a member of LTSE and can route orders to the exchange. As the customer, you're in charge of where your orders get routed, and you're welcome to specify a specific venue. Not every broker supports this right in their web UI, although I've heard that Interactive Brokers (IBKR) does.
If you've been following the news about controversies such as "pay for order flow" and such, you might make a connection here. Most retail customers don't specify where their order should be routed and as a result, they lend their voice to many of these industry practices. Because LTSE is not focused on maximizing trading volume, things are a little different on our platform. If you route to LTSE, you can be sure that no intermediary is getting paid for your transaction, and LTSE does not take a trading fee either.
As to whether traffic on LTSE helps, I think it does lend a little bit of credibility. If you take the time to tell your broker this is what you want, they will probably notice.
Thanks for the thoughtful question!