The Cardinals have never blamed the lack of a DH on their disinterest in giving out decade-plus deals on the free-agent market, at least not to me.
The best example of that reluctance, I think, comes from the Cardinals realizing they made the right call by not outbidding the Angels for Albert Pujols.
If you go through the pain of letting a generational talent walk because you can't justify the terms, and the long run proves you right, then you have some real hesitancy of giving out mega deals to free agents you did not draft, develop and win a World Series championship with.
As a separate topic, NL teams have been at a clear disadvantage to AL teams in terms of long-term contracts in the past because they could not shift aging players to the DH spots as their defense eroded. An evening of that playing field has always made sense, regardless of how people feel about the rule. It's crazy to me MLB has allowed teams to have such different rules in play for so long, this being one reason why.
I don't think the Cardinals are the team to bet on when it comes to winning free-agent bidding wars for top-of-the-line free agents, and I'll feel that way until I see proof otherwise. They have preferred to make their big-splash moves on trades, where they know the contract they are taking on and are negotiating the details with one team. It's a more controlled process and there is comfort in that. Also, success.